Daily Papers

This paper studies Kernel density estimation for a high-dimensional distribution rho(x). Traditional approaches have focused on the limit of large number of data points n and fixed dimension d. We analyze instead the regime where both the number n of data points y_i and their dimensionality d grow with a fixed ratio alpha=(log n)/d. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density hat rho_h^{D}(x)=1{n h^d}sum_{i=1}^n Kleft(x-y_i{h}right), depending on the bandwidth h: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, h_{CLT}(alpha), we find that the CLT breaks down. The statistics of hat rho_h^{D}(x) for a fixed x drawn from rho(x) is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value h_G(alpha), we find that hat rho_h^{D}(x) is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. Our findings reveal limitations of classical approaches, show the relevance of these new statistical regimes, and offer new insights for Kernel density estimation in high-dimensional settings.

We present TraDE, a self-attention-based architecture for auto-regressive density estimation with continuous and discrete valued data. Our model is trained using a penalized maximum likelihood objective, which ensures that samples from the density estimate resemble the training data distribution. The use of self-attention means that the model need not retain conditional sufficient statistics during the auto-regressive process beyond what is needed for each covariate. On standard tabular and image data benchmarks, TraDE produces significantly better density estimates than existing approaches such as normalizing flow estimators and recurrent auto-regressive models. However log-likelihood on held-out data only partially reflects how useful these estimates are in real-world applications. In order to systematically evaluate density estimators, we present a suite of tasks such as regression using generated samples, out-of-distribution detection, and robustness to noise in the training data and demonstrate that TraDE works well in these scenarios.

Autoregressive models are among the best performing neural density estimators. We describe an approach for increasing the flexibility of an autoregressive model, based on modelling the random numbers that the model uses internally when generating data. By constructing a stack of autoregressive models, each modelling the random numbers of the next model in the stack, we obtain a type of normalizing flow suitable for density estimation, which we call Masked Autoregressive Flow. This type of flow is closely related to Inverse Autoregressive Flow and is a generalization of Real NVP. Masked Autoregressive Flow achieves state-of-the-art performance in a range of general-purpose density estimation tasks.

Out-of-distribution (OOD) detection is a critical requirement for the deployment of deep neural networks. This paper introduces the HEAT model, a new post-hoc OOD detection method estimating the density of in-distribution (ID) samples using hybrid energy-based models (EBM) in the feature space of a pre-trained backbone. HEAT complements prior density estimators of the ID density, e.g. parametric models like the Gaussian Mixture Model (GMM), to provide an accurate yet robust density estimation. A second contribution is to leverage the EBM framework to provide a unified density estimation and to compose several energy terms. Extensive experiments demonstrate the significance of the two contributions. HEAT sets new state-of-the-art OOD detection results on the CIFAR-10 / CIFAR-100 benchmark as well as on the large-scale Imagenet benchmark. The code is available at: https://github.com/MarcLafon/heatood.

The field of multi-object tracking has recently seen a renewed interest in the good old schema of tracking-by-detection, as its simplicity and strong priors spare it from the complex design and painful babysitting of tracking-by-attention approaches. In view of this, we aim at extending tracking-by-detection to multi-modal settings, where a comprehensive cost has to be computed from heterogeneous information e.g., 2D motion cues, visual appearance, and pose estimates. More precisely, we follow a case study where a rough estimate of 3D information is also available and must be merged with other traditional metrics (e.g., the IoU). To achieve that, recent approaches resort to either simple rules or complex heuristics to balance the contribution of each cost. However, i) they require careful tuning of tailored hyperparameters on a hold-out set, and ii) they imply these costs to be independent, which does not hold in reality. We address these issues by building upon an elegant probabilistic formulation, which considers the cost of a candidate association as the negative log-likelihood yielded by a deep density estimator, trained to model the conditional joint probability distribution of correct associations. Our experiments, conducted on both simulated and real benchmarks, show that our approach consistently enhances the performance of several tracking-by-detection algorithms.

Simulation-based inference with conditional neural density estimators is a powerful approach to solving inverse problems in science. However, these methods typically treat the underlying forward model as a black box, with no way to exploit geometric properties such as equivariances. Equivariances are common in scientific models, however integrating them directly into expressive inference networks (such as normalizing flows) is not straightforward. We here describe an alternative method to incorporate equivariances under joint transformations of parameters and data. Our method -- called group equivariant neural posterior estimation (GNPE) -- is based on self-consistently standardizing the "pose" of the data while estimating the posterior over parameters. It is architecture-independent, and applies both to exact and approximate equivariances. As a real-world application, we use GNPE for amortized inference of astrophysical binary black hole systems from gravitational-wave observations. We show that GNPE achieves state-of-the-art accuracy while reducing inference times by three orders of magnitude.

MATSim (Multi-Agent Transport Simulation Toolkit) is an open source large-scale agent-based transportation planning project applied to various areas like road transport, public transport, freight transport, regional evacuation, etc. BEAM (Behavior, Energy, Autonomy, and Mobility) framework extends MATSim to enable powerful and scalable analysis of urban transportation systems. The agents from the BEAM simulation exhibit 'mode choice' behavior based on multinomial logit model. In our study, we consider eight mode choices viz. bike, car, walk, ride hail, driving to transit, walking to transit, ride hail to transit, and ride hail pooling. The 'alternative specific constants' for each mode choice are critical hyperparameters in a configuration file related to a particular scenario under experimentation. We use the 'Urbansim-10k' BEAM scenario (with 10,000 population size) for all our experiments. Since these hyperparameters affect the simulation in complex ways, manual calibration methods are time consuming. We present a parallel Bayesian optimization method with early stopping rule to achieve fast convergence for the given multi-in-multi-out problem to its optimal configurations. Our model is based on an open source HpBandSter package. This approach combines hierarchy of several 1D Kernel Density Estimators (KDE) with a cheap evaluator (Hyperband, a single multidimensional KDE). Our model has also incorporated extrapolation based early stopping rule. With our model, we could achieve a 25% L1 norm for a large-scale BEAM simulation in fully autonomous manner. To the best of our knowledge, our work is the first of its kind applied to large-scale multi-agent transportation simulations. This work can be useful for surrogate modeling of scenarios with very large populations.

Models for learning probability distributions such as generative models and density estimators behave quite differently from models for learning functions. One example is found in the memorization phenomenon, namely the ultimate convergence to the empirical distribution, that occurs in generative adversarial networks (GANs). For this reason, the issue of generalization is more subtle than that for supervised learning. For the bias potential model, we show that dimension-independent generalization accuracy is achievable if early stopping is adopted, despite that in the long term, the model either memorizes the samples or diverges.

Deep generative models have been demonstrated as state-of-the-art density estimators. Yet, recent work has found that they often assign a higher likelihood to data from outside the training distribution. This seemingly paradoxical behavior has caused concerns over the quality of the attained density estimates. In the context of hierarchical variational autoencoders, we provide evidence to explain this behavior by out-of-distribution data having in-distribution low-level features. We argue that this is both expected and desirable behavior. With this insight in hand, we develop a fast, scalable and fully unsupervised likelihood-ratio score for OOD detection that requires data to be in-distribution across all feature-levels. We benchmark the method on a vast set of data and model combinations and achieve state-of-the-art results on out-of-distribution detection.

We propose a physics-informed neural particle method (PINN--PM) for the spatially homogeneous Landau equation. The method adopts a Lagrangian interacting-particle formulation and jointly parameterizes the time-dependent score and the characteristic flow map with neural networks. Instead of advancing particles through explicit time stepping, the Landau dynamics is enforced via a continuous-time residual defined along particle trajectories. This design removes time-discretization error and yields a mesh-free solver that can be queried at arbitrary times without sequential integration. We establish a rigorous stability analysis in an L^2_v framework. The deviation between learned and exact characteristics is controlled by three interpretable sources: (i) score approximation error, (ii) empirical particle approximation error, and (iii) the physics residual of the neural flow. This trajectory estimate propagates to density reconstruction, where we derive an L^2_v error bound for kernel density estimators combining classical bias--variance terms with a trajectory-induced contribution. Using Hyvarinen's identity, we further relate the oracle score-matching gap to the L^2_v score error and show that the empirical loss concentrates at the Monte Carlo rate, yielding computable a posteriori accuracy certificates. Numerical experiments on analytical benchmarks, including the two- and three-dimensional BKW solutions, as well as reference-free configurations, demonstrate stable transport, preservation of macroscopic invariants, and competitive or improved accuracy compared with time-stepping score-based particle and blob methods while using significantly fewer particles.

Normalizing flows are powerful non-parametric statistical models that function as a hybrid between density estimators and generative models. Current learning algorithms for normalizing flows assume that data points are sampled independently, an assumption that is frequently violated in practice, which may lead to erroneous density estimation and data generation. We propose a likelihood objective of normalizing flows incorporating dependencies between the data points, for which we derive a flexible and efficient learning algorithm suitable for different dependency structures. We show that respecting dependencies between observations can improve empirical results on both synthetic and real-world data, and leads to higher statistical power in a downstream application to genome-wide association studies.

Once deployed, medical image analysis methods are often faced with unexpected image corruptions and noise perturbations. These unknown covariate shifts present significant challenges to deep learning based methods trained on "clean" images. This often results in unreliable predictions and poorly calibrated confidence, hence hindering clinical applicability. While recent methods have been developed to address specific issues such as confidence calibration or adversarial robustness, no single framework effectively tackles all these challenges simultaneously. To bridge this gap, we propose LaDiNE, a novel ensemble learning method combining the robustness of Vision Transformers with diffusion-based generative models for improved reliability in medical image classification. Specifically, transformer encoder blocks are used as hierarchical feature extractors that learn invariant features from images for each ensemble member, resulting in features that are robust to input perturbations. In addition, diffusion models are used as flexible density estimators to estimate member densities conditioned on the invariant features, leading to improved modeling of complex data distributions while retaining properly calibrated confidence. Extensive experiments on tuberculosis chest X-rays and melanoma skin cancer datasets demonstrate that LaDiNE achieves superior performance compared to a wide range of state-of-the-art methods by simultaneously improving prediction accuracy and confidence calibration under unseen noise, adversarial perturbations, and resolution degradation.

Albeit worryingly underrated in the recent literature on machine learning in general (and, on deep learning in particular), multivariate density estimation is a fundamental task in many applications, at least implicitly, and still an open issue. With a few exceptions, deep neural networks (DNNs) have seldom been applied to density estimation, mostly due to the unsupervised nature of the estimation task, and (especially) due to the need for constrained training algorithms that ended up realizing proper probabilistic models that satisfy Kolmogorov's axioms. Moreover, in spite of the well-known improvement in terms of modeling capabilities yielded by mixture models over plain single-density statistical estimators, no proper mixtures of multivariate DNN-based component densities have been investigated so far. The paper fills this gap by extending our previous work on Neural Mixture Densities (NMMs) to multivariate DNN mixtures. A maximum-likelihood (ML) algorithm for estimating Deep NMMs (DNMMs) is handed out, which satisfies numerically a combination of hard and soft constraints aimed at ensuring satisfaction of Kolmogorov's axioms. The class of probability density functions that can be modeled to any degree of precision via DNMMs is formally defined. A procedure for the automatic selection of the DNMM architecture, as well as of the hyperparameters for its ML training algorithm, is presented (exploiting the probabilistic nature of the DNMM). Experimental results on univariate and multivariate data are reported on, corroborating the effectiveness of the approach and its superiority to the most popular statistical estimation techniques.

Federated distillation has emerged as a promising collaborative machine learning approach, offering enhanced privacy protection and reduced communication compared to traditional federated learning by exchanging model outputs (soft logits) rather than full model parameters. However, existing methods employ complex selective knowledge-sharing strategies that require clients to identify in-distribution proxy data through computationally expensive statistical density ratio estimators. Additionally, server-side filtering of ambiguous knowledge introduces latency to the process. To address these challenges, we propose a robust, resource-efficient EdgeFD method that reduces the complexity of the client-side density ratio estimation and removes the need for server-side filtering. EdgeFD introduces an efficient KMeans-based density ratio estimator for effectively filtering both in-distribution and out-of-distribution proxy data on clients, significantly improving the quality of knowledge sharing. We evaluate EdgeFD across diverse practical scenarios, including strong non-IID, weak non-IID, and IID data distributions on clients, without requiring a pre-trained teacher model on the server for knowledge distillation. Experimental results demonstrate that EdgeFD outperforms state-of-the-art methods, consistently achieving accuracy levels close to IID scenarios even under heterogeneous and challenging conditions. The significantly reduced computational overhead of the KMeans-based estimator is suitable for deployment on resource-constrained edge devices, thereby enhancing the scalability and real-world applicability of federated distillation. The code is available online for reproducibility.

While efficient distribution learning is no doubt behind the groundbreaking success of diffusion modeling, its theoretical guarantees are quite limited. In this paper, we provide the first rigorous analysis on approximation and generalization abilities of diffusion modeling for well-known function spaces. The highlight of this paper is that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates in the total variation distance and in the Wasserstein distance of order one. Furthermore, we extend our theory to demonstrate how diffusion models adapt to low-dimensional data distributions. We expect these results advance theoretical understandings of diffusion modeling and its ability to generate verisimilar outputs.

Despite their impressive predictive performance in various computer vision tasks, deep neural networks (DNNs) tend to make overly confident predictions, which hinders their widespread use in safety-critical applications. While there have been recent attempts to calibrate DNNs, most of these efforts have primarily been focused on classification tasks, thus neglecting DNN-based object detectors. Although several recent works addressed calibration for object detection and proposed differentiable penalties, none of them are consistent estimators of established concepts in calibration. In this work, we tackle the challenge of defining and estimating calibration error specifically for this task. In particular, we adapt the definition of classification calibration error to handle the nuances associated with object detection, and predictions in structured output spaces more generally. Furthermore, we propose a consistent and differentiable estimator of the detection calibration error, utilizing kernel density estimation. Our experiments demonstrate the effectiveness of our estimator against competing train-time and post-hoc calibration methods, while maintaining similar detection performance.

Despite advances in depth estimation, flying points remain a persistent failure mode: near object boundaries, depth estimators often predict spurious 3D points in the empty space between foreground and background surfaces. We trace this artifact to a standard modeling choice: assigning each pixel a single depth hypothesis. At boundaries, a pixel can straddle a foreground and a background surface, so its true depth is ambiguous between the two. A model that predicts a single depth cannot keep both possibilities, so training instead pulls the prediction toward an intermediate depth that lies on neither surface. We address this with MDA, a mixture-density representation that lets the model predict multiple depth hypotheses and their associated probabilities for each pixel. Near boundaries, different hypotheses can align with different surfaces, and the decoded depth is selected from one of these hypotheses rather than placed in the empty space between them. Across different backbones, MDA substantially improves boundary reconstruction and largely removes flying-point artifacts even under severe input blur, while adding negligible runtime overhead. The same mixture-density framework naturally extends to transparent objects, where it predicts multiple depth layers at transparent pixels, and to sky regions, where a dedicated component separates the unbounded sky from finite-depth regions, producing flying-point-free skylines. Project Page: https://biansy000.github.io/mda-site/.

Learning unnormalized statistical models (e.g., energy-based models) is computationally challenging due to the complexity of handling the partition function. To eschew this complexity, noise-contrastive estimation~(NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. However, as found in previous works, NCE may perform poorly in many tasks due to its flat loss landscape and slow convergence. In this paper, we study it a direct approach for optimizing the negative log-likelihood of unnormalized models from the perspective of compositional optimization. To tackle the partition function, a noise distribution is introduced such that the log partition function can be written as a compositional function whose inner function can be estimated with stochastic samples. Hence, the objective can be optimized by stochastic compositional optimization algorithms. Despite being a simple method, we demonstrate that it is more favorable than NCE by (1) establishing a fast convergence rate and quantifying its dependence on the noise distribution through the variance of stochastic estimators; (2) developing better results for one-dimensional Gaussian mean estimation by showing our objective has a much favorable loss landscape and hence our method enjoys faster convergence; (3) demonstrating better performance on multiple applications, including density estimation, out-of-distribution detection, and real image generation.

The Cambrian explosion of easily accessible pre-trained diffusion models suggests a demand for methods that combine multiple different pre-trained diffusion models without incurring the significant computational burden of re-training a larger combined model. In this paper, we cast the problem of combining multiple pre-trained diffusion models at the generation stage under a novel proposed framework termed superposition. Theoretically, we derive superposition from rigorous first principles stemming from the celebrated continuity equation and design two novel algorithms tailor-made for combining diffusion models in SuperDiff. SuperDiff leverages a new scalable It\^o density estimator for the log likelihood of the diffusion SDE which incurs no additional overhead compared to the well-known Hutchinson's estimator needed for divergence calculations. We demonstrate that SuperDiff is scalable to large pre-trained diffusion models as superposition is performed solely through composition during inference, and also enjoys painless implementation as it combines different pre-trained vector fields through an automated re-weighting scheme. Notably, we show that SuperDiff is efficient during inference time, and mimics traditional composition operators such as the logical OR and the logical AND. We empirically demonstrate the utility of using SuperDiff for generating more diverse images on CIFAR-10, more faithful prompt conditioned image editing using Stable Diffusion, and improved unconditional de novo structure design of proteins. https://github.com/necludov/super-diffusion

The traditional kernel density estimator of an unknown density is by construction completely nonparametric, in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example the normal, while not losing much in precision when the true density is far from the parametric class. The idea is to multiply an initial parametric density estimate with a kernel type estimate of the necessary correction factor. This works well in cases where the correction factor function is less rough than the original density itself. Extensive comparisons with the kernel estimator are carried out, including exact analysis for the class of all normal mixtures. The new method, with a normal start, wins quite often, even in many cases where the true density is far from normal. Procedures for choosing the smoothing parameter of the estimator are also discussed. The new estimator should be particularly useful in higher dimensions, where the usual nonparametric methods have problems. The idea is also spelled out for nonparametric regression.

The Beta kernel estimator offers a theoretically superior alternative to the Gaussian kernel for unit interval data, eliminating boundary bias without requiring reflection or transformation. However, its adoption remains limited by the lack of a reliable bandwidth selector; practitioners currently rely on iterative optimization methods that are computationally expensive and prone to instability. We derive the ``Beta Reference Rule,'' a fast, closed-form bandwidth selector based on the unweighted Asymptotic Mean Integrated Squared Error (AMISE) of a beta reference distribution. To address boundary integrability issues, we introduce a principled heuristic for U-shaped and J-shaped distributions. By employing a method-of-moments approximation, we reduce the bandwidth selection complexity from iterative optimization to O(1). Extensive Monte Carlo simulations demonstrate that our rule matches the accuracy of numerical optimization while delivering a speedup of over 35,000 times. Real-world validation on socioeconomic data shows that it avoids the ``vanishing boundary'' and ``shoulder'' artifacts common to Gaussian-based methods. We provide a comprehensive, open-source Python package to facilitate the immediate adoption of the Beta kernel as a drop-in replacement for standard density estimation tools.

Training of the neural autoregressive density estimator (NADE) can be viewed as doing one step of probabilistic inference on missing values in data. We propose a new model that extends this inference scheme to multiple steps, arguing that it is easier to learn to improve a reconstruction in k steps rather than to learn to reconstruct in a single inference step. The proposed model is an unsupervised building block for deep learning that combines the desirable properties of NADE and multi-predictive training: (1) Its test likelihood can be computed analytically, (2) it is easy to generate independent samples from it, and (3) it uses an inference engine that is a superset of variational inference for Boltzmann machines. The proposed NADE-k is competitive with the state-of-the-art in density estimation on the two datasets tested.

Existing machine learning methods for causal inference usually estimate quantities expressed via the mean of potential outcomes (e.g., average treatment effect). However, such quantities do not capture the full information about the distribution of potential outcomes. In this work, we estimate the density of potential outcomes after interventions from observational data. For this, we propose a novel, fully-parametric deep learning method called Interventional Normalizing Flows. Specifically, we combine two normalizing flows, namely (i) a nuisance flow for estimating nuisance parameters and (ii) a target flow for parametric estimation of the density of potential outcomes. We further develop a tractable optimization objective based on a one-step bias correction for efficient and doubly robust estimation of the target flow parameters. As a result, our Interventional Normalizing Flows offer a properly normalized density estimator. Across various experiments, we demonstrate that our Interventional Normalizing Flows are expressive and highly effective, and scale well with both sample size and high-dimensional confounding. To the best of our knowledge, our Interventional Normalizing Flows are the first proper fully-parametric, deep learning method for density estimation of potential outcomes.

A prominent family of methods for learning data distributions relies on density ratio estimation (DRE), where a model is trained to classify between data samples and samples from some reference distribution. DRE-based models can directly output the likelihood for any given input, a highly desired property that is lacking in most generative techniques. Nevertheless, to date, DRE methods have failed in accurately capturing the distributions of complex high-dimensional data, like images, and have thus been drawing reduced research attention in recent years. In this work we present classification diffusion models (CDMs), a DRE-based generative method that adopts the formalism of denoising diffusion models (DDMs) while making use of a classifier that predicts the level of noise added to a clean signal. Our method is based on an analytical connection that we derive between the MSE-optimal denoiser for removing white Gaussian noise and the cross-entropy-optimal classifier for predicting the noise level. Our method is the first DRE-based technique that can successfully generate images beyond the MNIST dataset. Furthermore, it can output the likelihood of any input in a single forward pass, achieving state-of-the-art negative log likelihood (NLL) among methods with this property. Code is available on the project's webpage in https://shaharYadin.github.io/CDM/ .

We propose a direct estimation method for R\'{e}nyi and f-divergence measures based on a new graph theoretical interpretation. Suppose that we are given two sample sets X and Y, respectively with N and M samples, where eta:=M/N is a constant value. Considering the k-nearest neighbor (k-NN) graph of Y in the joint data set (X,Y), we show that the average powered ratio of the number of X points to the number of Y points among all k-NN points is proportional to R\'{e}nyi divergence of X and Y densities. A similar method can also be used to estimate f-divergence measures. We derive bias and variance rates, and show that for the class of gamma-H\"{o}lder smooth functions, the estimator achieves the MSE rate of O(N^{-2gamma/(gamma+d)}). Furthermore, by using a weighted ensemble estimation technique, for density functions with continuous and bounded derivatives of up to the order d, and some extra conditions at the support set boundary, we derive an ensemble estimator that achieves the parametric MSE rate of O(1/N). Our estimators are more computationally tractable than other competing estimators, which makes them appealing in many practical applications.

Identifying Out-of-distribution (OOD) data is becoming increasingly critical as the real-world applications of deep learning methods expand. Post-hoc methods modify softmax scores fine-tuned on outlier data or leverage intermediate feature layers to identify distinctive patterns between In-Distribution (ID) and OOD samples. Other methods focus on employing diverse OOD samples to learn discrepancies between ID and OOD. These techniques, however, are typically dependent on the quality of the outlier samples assumed. Density-based methods explicitly model class-conditioned distributions but this requires long training time or retraining the classifier. To tackle these issues, we introduce FlowCon, a new density-based OOD detection technique. Our main innovation lies in efficiently combining the properties of normalizing flow with supervised contrastive learning, ensuring robust representation learning with tractable density estimation. Empirical evaluation shows the enhanced performance of our method across common vision datasets such as CIFAR-10 and CIFAR-100 pretrained on ResNet18 and WideResNet classifiers. We also perform quantitative analysis using likelihood plots and qualitative visualization using UMAP embeddings and demonstrate the robustness of the proposed method under various OOD contexts. Code will be open-sourced post decision.

We study nonparametric density estimation in non-stationary drift settings. Given a sequence of independent samples taken from a distribution that gradually changes in time, the goal is to compute the best estimate for the current distribution. We prove tight minimax risk bounds for both discrete and continuous smooth densities, where the minimum is over all possible estimates and the maximum is over all possible distributions that satisfy the drift constraints. Our technique handles a broad class of drift models, and generalizes previous results on agnostic learning under drift.

While previous distribution shift detection approaches can identify if a shift has occurred, these approaches cannot localize which specific features have caused a distribution shift -- a critical step in diagnosing or fixing any underlying issue. For example, in military sensor networks, users will want to detect when one or more of the sensors has been compromised, and critically, they will want to know which specific sensors might be compromised. Thus, we first define a formalization of this problem as multiple conditional distribution hypothesis tests and propose both non-parametric and parametric statistical tests. For both efficiency and flexibility, we then propose to use a test statistic based on the density model score function (i.e. gradient with respect to the input) -- which can easily compute test statistics for all dimensions in a single forward and backward pass. Any density model could be used for computing the necessary statistics including deep density models such as normalizing flows or autoregressive models. We additionally develop methods for identifying when and where a shift occurs in multivariate time-series data and show results for multiple scenarios using realistic attack models on both simulated and real world data.

Policy constraint methods to offline reinforcement learning (RL) typically utilize parameterization or regularization that constrains the policy to perform actions within the support set of the behavior policy. The elaborative designs of parameterization methods usually intrude into the policy networks, which may bring extra inference cost and cannot take full advantage of well-established online methods. Regularization methods reduce the divergence between the learned policy and the behavior policy, which may mismatch the inherent density-based definition of support set thereby failing to avoid the out-of-distribution actions effectively. This paper presents Supported Policy OpTimization (SPOT), which is directly derived from the theoretical formalization of the density-based support constraint. SPOT adopts a VAE-based density estimator to explicitly model the support set of behavior policy and presents a simple but effective density-based regularization term, which can be plugged non-intrusively into off-the-shelf off-policy RL algorithms. SPOT achieves the state-of-the-art performance on standard benchmarks for offline RL. Benefiting from the pluggable design, offline pretrained models from SPOT can also be applied to perform online fine-tuning seamlessly.

Sequential Bayesian inference can be used for continual learning to prevent catastrophic forgetting of past tasks and provide an informative prior when learning new tasks. We revisit sequential Bayesian inference and test whether having access to the true posterior is guaranteed to prevent catastrophic forgetting in Bayesian neural networks. To do this we perform sequential Bayesian inference using Hamiltonian Monte Carlo. We propagate the posterior as a prior for new tasks by fitting a density estimator on Hamiltonian Monte Carlo samples. We find that this approach fails to prevent catastrophic forgetting demonstrating the difficulty in performing sequential Bayesian inference in neural networks. From there we study simple analytical examples of sequential Bayesian inference and CL and highlight the issue of model misspecification which can lead to sub-optimal continual learning performance despite exact inference. Furthermore, we discuss how task data imbalances can cause forgetting. From these limitations, we argue that we need probabilistic models of the continual learning generative process rather than relying on sequential Bayesian inference over Bayesian neural network weights. In this vein, we also propose a simple baseline called Prototypical Bayesian Continual Learning, which is competitive with state-of-the-art Bayesian continual learning methods on class incremental continual learning vision benchmarks.

Offline reinforcement learning (RL) provides a promising solution to learning an agent fully relying on a data-driven paradigm. However, constrained by the limited quality of the offline dataset, its performance is often sub-optimal. Therefore, it is desired to further finetune the agent via extra online interactions before deployment. Unfortunately, offline-to-online RL can be challenging due to two main challenges: constrained exploratory behavior and state-action distribution shift. To this end, we propose a Simple Unified uNcertainty-Guided (SUNG) framework, which naturally unifies the solution to both challenges with the tool of uncertainty. Specifically, SUNG quantifies uncertainty via a VAE-based state-action visitation density estimator. To facilitate efficient exploration, SUNG presents a practical optimistic exploration strategy to select informative actions with both high value and high uncertainty. Moreover, SUNG develops an adaptive exploitation method by applying conservative offline RL objectives to high-uncertainty samples and standard online RL objectives to low-uncertainty samples to smoothly bridge offline and online stages. SUNG achieves state-of-the-art online finetuning performance when combined with different offline RL methods, across various environments and datasets in D4RL benchmark.

Bayesian inference usually requires running potentially costly inference procedures separately for every new observation. In contrast, the idea of amortized Bayesian inference is to initially invest computational cost in training an inference network on simulated data, which can subsequently be used to rapidly perform inference (i.e., to return estimates of posterior distributions) for new observations. This approach has been applied to many real-world models in the sciences and engineering, but it is unclear how robust the approach is to adversarial perturbations in the observed data. Here, we study the adversarial robustness of amortized Bayesian inference, focusing on simulation-based estimation of multi-dimensional posterior distributions. We show that almost unrecognizable, targeted perturbations of the observations can lead to drastic changes in the predicted posterior and highly unrealistic posterior predictive samples, across several benchmark tasks and a real-world example from neuroscience. We propose a computationally efficient regularization scheme based on penalizing the Fisher information of the conditional density estimator, and show how it improves the adversarial robustness of amortized Bayesian inference.

Ensembles of independently trained neural networks are a state-of-the-art approach to estimate predictive uncertainty in Deep Learning, and can be interpreted as an approximation of the posterior distribution via a mixture of delta functions. The training of ensembles relies on non-convexity of the loss landscape and random initialization of their individual members, making the resulting posterior approximation uncontrolled. This paper proposes a novel and principled method to tackle this limitation, minimizing an f-divergence between the true posterior and a kernel density estimator (KDE) in a function space. We analyze this objective from a combinatorial point of view, and show that it is submodular with respect to mixture components for any f. Subsequently, we consider the problem of greedy ensemble construction. From the marginal gain on the negative f-divergence, which quantifies an improvement in posterior approximation yielded by adding a new component into the KDE, we derive a novel diversity term for ensemble methods. The performance of our approach is demonstrated on computer vision out-of-distribution detection benchmarks in a range of architectures trained on multiple datasets. The source code of our method is made publicly available at https://github.com/Oulu-IMEDS/greedy_ensembles_training.

Normalizing flows (NFs) have become a prominent method for deep generative models that allow for an analytic probability density estimation and efficient synthesis. However, a flow-based network is considered to be inefficient in parameter complexity because of reduced expressiveness of bijective mapping, which renders the models unfeasibly expensive in terms of parameters. We present an alternative parameterization scheme called NanoFlow, which uses a single neural density estimator to model multiple transformation stages. Hence, we propose an efficient parameter decomposition method and the concept of flow indication embedding, which are key missing components that enable density estimation from a single neural network. Experiments performed on audio and image models confirm that our method provides a new parameter-efficient solution for scalable NFs with significant sublinear parameter complexity.

Humans have a powerful and mysterious capacity to reason. By working through a series of purely mental steps, we can make inferences we would not be capable of making directly -- despite the fact that we get no additional data from the world. Similarly, when large language models generate a series of intermediate steps (a chain of thought) before answering a question, they often produce better answers than they otherwise would. We investigate why and how chain-of-thought reasoning is useful in language models, testing the hypothesis that reasoning is effective when training data consists of local clusters of variables that influence each other strongly. These training conditions enable the chaining of accurate local inferences in order to estimate relationships between variables that were not seen together in training. We prove that there will exist a "reasoning gap", where reasoning through intermediate variables improves inference, for the simple case of an autoregressive density estimator trained on local samples from a chain-structured probabilistic model. We then test our hypothesis empirically in more complex models, training an autoregressive language model on samples from Bayes nets but only including a subset of variables in each sample. We test language models' ability to match conditional probabilities with and without intermediate reasoning steps, finding that intermediate steps are only helpful when the training data is locally structured with respect to dependencies between variables and that the combination of locally-structured observations and reasoning is much more data-efficient than training on all variables. Our results illustrate how the effectiveness of reasoning step by step is rooted in the local statistical structure of the training data.

The theories of offline and online reinforcement learning, despite having evolved in parallel, have begun to show signs of the possibility for a unification, with algorithms and analysis techniques for one setting often having natural counterparts in the other. However, the notion of density ratio modeling, an emerging paradigm in offline RL, has been largely absent from online RL, perhaps for good reason: the very existence and boundedness of density ratios relies on access to an exploratory dataset with good coverage, but the core challenge in online RL is to collect such a dataset without having one to start. In this work we show -- perhaps surprisingly -- that density ratio-based algorithms have online counterparts. Assuming only the existence of an exploratory distribution with good coverage, a structural condition known as coverability (Xie et al., 2023), we give a new algorithm (GLOW) that uses density ratio realizability and value function realizability to perform sample-efficient online exploration. GLOW addresses unbounded density ratios via careful use of truncation, and combines this with optimism to guide exploration. GLOW is computationally inefficient; we complement it with a more efficient counterpart, HyGLOW, for the Hybrid RL setting (Song et al., 2022) wherein online RL is augmented with additional offline data. HyGLOW is derived as a special case of a more general meta-algorithm that provides a provable black-box reduction from hybrid RL to offline RL, which may be of independent interest.

Density ratio estimation is fundamental to tasks involving f-divergences, yet existing methods often fail under significantly different distributions or inadequately overlapping supports -- the density-chasm and the support-chasm problems. Additionally, prior approaches yield divergent time scores near boundaries, leading to instability. We design D^3RE, a unified framework for robust, stable and efficient density ratio estimation. We propose the dequantified diffusion bridge interpolant (DDBI), which expands support coverage and stabilizes time scores via diffusion bridges and Gaussian dequantization. Building on DDBI, the proposed dequantified Schr{\"o}dinger bridge interpolant (DSBI) incorporates optimal transport to solve the Schr{\"o}dinger bridge problem, enhancing accuracy and efficiency. Our method offers uniform approximation and bounded time scores in theory, and outperforms baselines empirically in mutual information and density estimation tasks.

Bayesian optimization has attracted huge attention from diverse research areas in science and engineering, since it is capable of efficiently finding a global optimum of an expensive-to-evaluate black-box function. In general, a probabilistic regression model is widely used as a surrogate function to model an explicit distribution over function evaluations given an input to estimate and a training dataset. Beyond the probabilistic regression-based methods, density ratio estimation-based Bayesian optimization has been suggested in order to estimate a density ratio of the groups relatively close and relatively far to a global optimum. Developing this line of research further, supervised classifiers are employed to estimate a class probability for the two groups instead of a density ratio. However, the supervised classifiers used in this strategy are prone to be overconfident for known knowledge on global solution candidates. Supposing that we have access to unlabeled points, e.g., predefined fixed-size pools, we propose density ratio estimation-based Bayesian optimization with semi-supervised learning to solve this challenge. Finally, we show the empirical results of our methods and several baseline methods in two distinct scenarios with unlabeled point sampling and a fixed-size pool, and analyze the validity of our methods in diverse experiments.

In this research. we analyze the potential of Feature Density (HD) as a way to comparatively estimate machine learning (ML) classifier performance prior to training. The goal of the study is to aid in solving the problem of resource-intensive training of ML models which is becoming a serious issue due to continuously increasing dataset sizes and the ever rising popularity of Deep Neural Networks (DNN). The issue of constantly increasing demands for more powerful computational resources is also affecting the environment, as training large-scale ML models are causing alarmingly-growing amounts of CO2, emissions. Our approach 1s to optimize the resource-intensive training of ML models for Natural Language Processing to reduce the number of required experiments iterations. We expand on previous attempts on improving classifier training efficiency with FD while also providing an insight to the effectiveness of various linguistically-backed feature preprocessing methods for dialog classification, specifically cyberbullying detection.

Breast density estimation is one of the key tasks in recognizing individuals predisposed to breast cancer. It is often challenging because of low contrast and fluctuations in mammograms' fatty tissue background. Most of the time, the breast density is estimated manually where a radiologist assigns one of the four density categories decided by the Breast Imaging and Reporting Data Systems (BI-RADS). There have been efforts in the direction of automating a breast density classification pipeline. Breast density estimation is one of the key tasks performed during a screening exam. Dense breasts are more susceptible to breast cancer. The density estimation is challenging because of low contrast and fluctuations in mammograms' fatty tissue background. Traditional mammograms are being replaced by tomosynthesis and its other low radiation dose variants (for example Hologic' Intelligent 2D and C-View). Because of the low-dose requirement, increasingly more screening centers are favoring the Intelligent 2D view and C-View. Deep-learning studies for breast density estimation use only a single modality for training a neural network. However, doing so restricts the number of images in the dataset. In this paper, we show that a neural network trained on all the modalities at once performs better than a neural network trained on any single modality. We discuss these results using the area under the receiver operator characteristics curves.

The dynamic imbalance of the fore-background is a major challenge in video object counting, which is usually caused by the sparsity of target objects. This remains understudied in existing works and often leads to severe under-/over-prediction errors. To tackle this issue in video object counting, we propose a density-embedded Efficient Masked Autoencoder Counting (E-MAC) framework in this paper. To empower the model's representation ability on density regression, we develop a new Density-Embedded Masked mOdeling (DEMO) method, which first takes the density map as an auxiliary modality to perform multimodal self-representation learning for image and density map. Although DEMO contributes to effective cross-modal regression guidance, it also brings in redundant background information, making it difficult to focus on the foreground regions. To handle this dilemma, we propose an efficient spatial adaptive masking derived from density maps to boost efficiency. Meanwhile, we employ an optical flow-based temporal collaborative fusion strategy to effectively capture the dynamic variations across frames, aligning features to derive multi-frame density residuals. The counting accuracy of the current frame is boosted by harnessing the information from adjacent frames. In addition, considering that most existing datasets are limited to human-centric scenarios, we first propose a large video bird counting dataset, DroneBird, in natural scenarios for migratory bird protection. Extensive experiments on three crowd datasets and our DroneBird validate our superiority against the counterparts. The code and dataset are available.

Density of the reachable states can help understand the risk of safety-critical systems, especially in situations when worst-case reachability is too conservative. Recent work provides a data-driven approach to compute the density distribution of autonomous systems' forward reachable states online. In this paper, we study the use of such approach in combination with model predictive control for verifiable safe path planning under uncertainties. We first use the learned density distribution to compute the risk of collision online. If such risk exceeds the acceptable threshold, our method will plan for a new path around the previous trajectory, with the risk of collision below the threshold. Our method is well-suited to handle systems with uncertainties and complicated dynamics as our data-driven approach does not need an analytical form of the systems' dynamics and can estimate forward state density with an arbitrary initial distribution of uncertainties. We design two challenging scenarios (autonomous driving and hovercraft control) for safe motion planning in environments with obstacles under system uncertainties. We first show that our density estimation approach can reach a similar accuracy as the Monte-Carlo-based method while using only 0.01X training samples. By leveraging the estimated risk, our algorithm achieves the highest success rate in goal reaching when enforcing the safety rate above 0.99.

Estimating truncated density models is difficult, as these models have intractable normalising constants and hard to satisfy boundary conditions. Score matching can be adapted to solve the truncated density estimation problem, but requires a continuous weighting function which takes zero at the boundary and is positive elsewhere. Evaluation of such a weighting function (and its gradient) often requires a closed-form expression of the truncation boundary and finding a solution to a complicated optimisation problem. In this paper, we propose approximate Stein classes, which in turn leads to a relaxed Stein identity for truncated density estimation. We develop a novel discrepancy measure, truncated kernelised Stein discrepancy (TKSD), which does not require fixing a weighting function in advance, and can be evaluated using only samples on the boundary. We estimate a truncated density model by minimising the Lagrangian dual of TKSD. Finally, experiments show the accuracy of our method to be an improvement over previous works even without the explicit functional form of the boundary.

Flow-based generative models parameterize probability distributions through an invertible transformation and can be trained by maximum likelihood. Invertible residual networks provide a flexible family of transformations where only Lipschitz conditions rather than strict architectural constraints are needed for enforcing invertibility. However, prior work trained invertible residual networks for density estimation by relying on biased log-density estimates whose bias increased with the network's expressiveness. We give a tractable unbiased estimate of the log density using a "Russian roulette" estimator, and reduce the memory required during training by using an alternative infinite series for the gradient. Furthermore, we improve invertible residual blocks by proposing the use of activation functions that avoid derivative saturation and generalizing the Lipschitz condition to induced mixed norms. The resulting approach, called Residual Flows, achieves state-of-the-art performance on density estimation amongst flow-based models, and outperforms networks that use coupling blocks at joint generative and discriminative modeling.

Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-m rate, where m is the number of simulated samples. This can lead to significant computational challenges since a large m is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.

State density distribution, in contrast to worst-case reachability, can be leveraged for safety-related problems to better quantify the likelihood of the risk for potentially hazardous situations. In this work, we propose a data-driven method to compute the density distribution of reachable states for nonlinear and even black-box systems. Our semi-supervised approach learns system dynamics and the state density jointly from trajectory data, guided by the fact that the state density evolution follows the Liouville partial differential equation. With the help of neural network reachability tools, our approach can estimate the set of all possible future states as well as their density. Moreover, we could perform online safety verification with probability ranges for unsafe behaviors to occur. We use an extensive set of experiments to show that our learned solution can produce a much more accurate estimate on density distribution, and can quantify risks less conservatively and flexibly comparing with worst-case analysis.

We present Mantis Shrimp, a multi-survey deep learning model for photometric redshift estimation that fuses ultra-violet (GALEX), optical (PanSTARRS), and infrared (UnWISE) imagery. Machine learning is now an established approach for photometric redshift estimation, with generally acknowledged higher performance in areas with a high density of spectroscopically identified galaxies over template-based methods. Multiple works have shown that image-based convolutional neural networks can outperform tabular-based color/magnitude models. In comparison to tabular models, image models have additional design complexities: it is largely unknown how to fuse inputs from different instruments which have different resolutions or noise properties. The Mantis Shrimp model estimates the conditional density estimate of redshift using cutout images. The density estimates are well calibrated and the point estimates perform well in the distribution of available spectroscopically confirmed galaxies with (bias = 1e-2), scatter (NMAD = 2.44e-2) and catastrophic outlier rate (eta=17.53%). We find that early fusion approaches (e.g., resampling and stacking images from different instruments) match the performance of late fusion approaches (e.g., concatenating latent space representations), so that the design choice ultimately is left to the user. Finally, we study how the models learn to use information across bands, finding evidence that our models successfully incorporates information from all surveys. The applicability of our model to the analysis of large populations of galaxies is limited by the speed of downloading cutouts from external servers; however, our model could be useful in smaller studies such as generating priors over redshift for stellar population synthesis.

A promising class of generative models maps points from a simple distribution to a complex distribution through an invertible neural network. Likelihood-based training of these models requires restricting their architectures to allow cheap computation of Jacobian determinants. Alternatively, the Jacobian trace can be used if the transformation is specified by an ordinary differential equation. In this paper, we use Hutchinson's trace estimator to give a scalable unbiased estimate of the log-density. The result is a continuous-time invertible generative model with unbiased density estimation and one-pass sampling, while allowing unrestricted neural network architectures. We demonstrate our approach on high-dimensional density estimation, image generation, and variational inference, achieving the state-of-the-art among exact likelihood methods with efficient sampling.

In reinforcement learning and imitation learning, an object of central importance is the state distribution induced by the policy. It plays a crucial role in the policy gradient theorem, and references to it--along with the related state-action distribution--can be found all across the literature. Despite its importance, the state distribution is mostly discussed indirectly and theoretically, rather than being modeled explicitly. The reason being an absence of appropriate density estimation tools. In this work, we investigate applications of a normalizing flow-based model for the aforementioned distributions. In particular, we use a pair of flows coupled through the optimality point of the Donsker-Varadhan representation of the Kullback-Leibler (KL) divergence, for distribution matching based imitation learning. Our algorithm, Coupled Flow Imitation Learning (CFIL), achieves state-of-the-art performance on benchmark tasks with a single expert trajectory and extends naturally to a variety of other settings, including the subsampled and state-only regimes.

Offline reinforcement learning (RL) offers a promising direction for learning policies from pre-collected datasets without requiring further interactions with the environment. However, existing methods struggle to handle out-of-distribution (OOD) extrapolation errors, especially in sparse reward or scarce data settings. In this paper, we propose a novel training algorithm called Conservative Density Estimation (CDE), which addresses this challenge by explicitly imposing constraints on the state-action occupancy stationary distribution. CDE overcomes the limitations of existing approaches, such as the stationary distribution correction method, by addressing the support mismatch issue in marginal importance sampling. Our method achieves state-of-the-art performance on the D4RL benchmark. Notably, CDE consistently outperforms baselines in challenging tasks with sparse rewards or insufficient data, demonstrating the advantages of our approach in addressing the extrapolation error problem in offline RL.

A framework is proposed that addresses both conditional density estimation and latent variable discovery. The objective function maximizes explanation of variability in the data, achieved through the optimal transport barycenter generalized to a collection of conditional distributions indexed by a covariate --either given or latent-- in any suitable space. Theoretical results establish the existence of barycenters, a minimax formulation of optimal transport maps, and a general characterization of variability via the optimal transport cost. This framework leads to a family of non-parametric neural network-based algorithms, the BaryNet, with a supervised version that estimates conditional distributions and an unsupervised version that assigns latent variables. The efficacy of BaryNets is demonstrated by tests on both artificial and real-world data sets. A parallel drawn between autoencoders and the barycenter framework leads to the Barycentric autoencoder algorithm (BAE).

Post-hoc out-of-distribution (OOD) detection has garnered intensive attention in reliable machine learning. Many efforts have been dedicated to deriving score functions based on logits, distances, or rigorous data distribution assumptions to identify low-scoring OOD samples. Nevertheless, these estimate scores may fail to accurately reflect the true data density or impose impractical constraints. To provide a unified perspective on density-based score design, we propose a novel theoretical framework grounded in Bregman divergence, which extends distribution considerations to encompass an exponential family of distributions. Leveraging the conjugation constraint revealed in our theorem, we introduce a ConjNorm method, reframing density function design as a search for the optimal norm coefficient p against the given dataset. In light of the computational challenges of normalization, we devise an unbiased and analytically tractable estimator of the partition function using the Monte Carlo-based importance sampling technique. Extensive experiments across OOD detection benchmarks empirically demonstrate that our proposed ConjNorm has established a new state-of-the-art in a variety of OOD detection setups, outperforming the current best method by up to 13.25% and 28.19% (FPR95) on CIFAR-100 and ImageNet-1K, respectively.

This work studies the estimation of many statistical quantiles under differential privacy. More precisely, given a distribution and access to i.i.d. samples from it, we study the estimation of the inverse of its cumulative distribution function (the quantile function) at specific points. For instance, this task is of key importance in private data generation. We present two different approaches. The first one consists in privately estimating the empirical quantiles of the samples and using this result as an estimator of the quantiles of the distribution. In particular, we study the statistical properties of the recently published algorithm introduced by Kaplan et al. 2022 that privately estimates the quantiles recursively. The second approach is to use techniques of density estimation in order to uniformly estimate the quantile function on an interval. In particular, we show that there is a tradeoff between the two methods. When we want to estimate many quantiles, it is better to estimate the density rather than estimating the quantile function at specific points.

The modeling of probability distributions, specifically generative modeling and density estimation, has become an immensely popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and texts. Nevertheless, a theoretical understanding of its success is still incomplete. One mystery is the paradox between memorization and generalization: In theory, the model is trained to be exactly the same as the empirical distribution of the finite samples, whereas in practice, the trained model can generate new samples or estimate the likelihood of unseen samples. Likewise, the overwhelming diversity of distribution learning models calls for a unified perspective on this subject. This paper provides a mathematical framework such that all the well-known models can be derived based on simple principles. To demonstrate its efficacy, we present a survey of our results on the approximation error, training error and generalization error of these models, which can all be established based on this framework. In particular, the aforementioned paradox is resolved by proving that these models enjoy implicit regularization during training, so that the generalization error at early-stopping avoids the curse of dimensionality. Furthermore, we provide some new results on landscape analysis and the mode collapse phenomenon.

Diffusion models have emerged as a powerful class of generative models, capable of producing high-quality images by mapping noise to a data distribution. However, recent findings suggest that image likelihood does not align with perceptual quality: high-likelihood samples tend to be smooth, while lower-likelihood ones are more detailed. Controlling sample density is thus crucial for balancing realism and detail. In this paper, we analyze an existing technique, Prior Guidance, which scales the latent code to influence image detail. We introduce score alignment, a condition that explains why this method works and show that it can be tractably checked for any continuous normalizing flow model. We then propose Density Guidance, a principled modification of the generative ODE that enables exact log-density control during sampling. Finally, we extend Density Guidance to stochastic sampling, ensuring precise log-density control while allowing controlled variation in structure or fine details. Our experiments demonstrate that these techniques provide fine-grained control over image detail without compromising sample quality.

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to couple the base and the target densities. This enables us to incorporate information about class labels or continuous embeddings to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting.

Volumetry is one of the principal downstream applications of 3D medical image segmentation, for example, to detect abnormal tissue growth or for surgery planning. Conformal Prediction is a promising framework for uncertainty quantification, providing calibrated predictive intervals associated with automatic volume measurements. However, this methodology is based on the hypothesis that calibration and test samples are exchangeable, an assumption that is in practice often violated in medical image applications. A weighted formulation of Conformal Prediction can be framed to mitigate this issue, but its empirical investigation in the medical domain is still lacking. A potential reason is that it relies on the estimation of the density ratio between the calibration and test distributions, which is likely to be intractable in scenarios involving high-dimensional data. To circumvent this, we propose an efficient approach for density ratio estimation relying on the compressed latent representations generated by the segmentation model. Our experiments demonstrate the efficiency of our approach to reduce the coverage error in the presence of covariate shifts, in both synthetic and real-world settings. Our implementation is available at https://github.com/benolmbrt/wcp_miccai

Known for their impressive performance in generative modeling, diffusion models are attractive candidates for density-based anomaly detection. This paper investigates different variations of diffusion modeling for unsupervised and semi-supervised anomaly detection. In particular, we find that Denoising Diffusion Probability Models (DDPM) are performant on anomaly detection benchmarks yet computationally expensive. By simplifying DDPM in application to anomaly detection, we are naturally led to an alternative approach called Diffusion Time Estimation (DTE). DTE estimates the distribution over diffusion time for a given input and uses the mode or mean of this distribution as the anomaly score. We derive an analytical form for this density and leverage a deep neural network to improve inference efficiency. Through empirical evaluations on the ADBench benchmark, we demonstrate that all diffusion-based anomaly detection methods perform competitively for both semi-supervised and unsupervised settings. Notably, DTE achieves orders of magnitude faster inference time than DDPM, while outperforming it on this benchmark. These results establish diffusion-based anomaly detection as a scalable alternative to traditional methods and recent deep-learning techniques for standard unsupervised and semi-supervised anomaly detection settings.

Neural volume rendering became increasingly popular recently due to its success in synthesizing novel views of a scene from a sparse set of input images. So far, the geometry learned by neural volume rendering techniques was modeled using a generic density function. Furthermore, the geometry itself was extracted using an arbitrary level set of the density function leading to a noisy, often low fidelity reconstruction. The goal of this paper is to improve geometry representation and reconstruction in neural volume rendering. We achieve that by modeling the volume density as a function of the geometry. This is in contrast to previous work modeling the geometry as a function of the volume density. In more detail, we define the volume density function as Laplace's cumulative distribution function (CDF) applied to a signed distance function (SDF) representation. This simple density representation has three benefits: (i) it provides a useful inductive bias to the geometry learned in the neural volume rendering process; (ii) it facilitates a bound on the opacity approximation error, leading to an accurate sampling of the viewing ray. Accurate sampling is important to provide a precise coupling of geometry and radiance; and (iii) it allows efficient unsupervised disentanglement of shape and appearance in volume rendering. Applying this new density representation to challenging scene multiview datasets produced high quality geometry reconstructions, outperforming relevant baselines. Furthermore, switching shape and appearance between scenes is possible due to the disentanglement of the two.

Random forests is a common non-parametric regression technique which performs well for mixed-type data and irrelevant covariates, while being robust to monotonic variable transformations. Existing random forest implementations target regression or classification. We introduce the RFCDE package for fitting random forest models optimized for nonparametric conditional density estimation, including joint densities for multiple responses. This enables analysis of conditional probability distributions which is useful for propagating uncertainty and of joint distributions that describe relationships between multiple responses and covariates. RFCDE is released under the MIT open-source license and can be accessed at https://github.com/tpospisi/rfcde . Both R and Python versions, which call a common C++ library, are available.

Recent work has shown how denoising and contractive autoencoders implicitly capture the structure of the data-generating density, in the case where the corruption noise is Gaussian, the reconstruction error is the squared error, and the data is continuous-valued. This has led to various proposals for sampling from this implicitly learned density function, using Langevin and Metropolis-Hastings MCMC. However, it remained unclear how to connect the training procedure of regularized auto-encoders to the implicit estimation of the underlying data-generating distribution when the data are discrete, or using other forms of corruption process and reconstruction errors. Another issue is the mathematical justification which is only valid in the limit of small corruption noise. We propose here a different attack on the problem, which deals with all these issues: arbitrary (but noisy enough) corruption, arbitrary reconstruction loss (seen as a log-likelihood), handling both discrete and continuous-valued variables, and removing the bias due to non-infinitesimal corruption noise (or non-infinitesimal contractive penalty).

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

In astrophysical and cosmological analyses, the increasing quality and volume of astronomical data demand efficient and precise computational tools. This work introduces a novel adaptive algorithm for automatic knots (AutoKnots) allocation in spline interpolation, designed to meet user-defined precision requirements. Unlike traditional methods that rely on manually configured knot distributions with numerous parameters, the proposed technique automatically determines the optimal number and placement of knots based on interpolation error criteria. This simplifies configuration, often requiring only a single parameter. The algorithm progressively improves the interpolation by adaptively sampling the function-to-be-approximated, f(x), in regions where the interpolation error exceeds the desired threshold. All function evaluations contribute directly to the final approximation, ensuring efficiency. While each resampling step involves recomputing the interpolation table, this process is highly optimized and usually computationally negligible compared to the cost of evaluating f(x). We show the algorithm's efficacy through a series of precision tests on different functions. However, the study underscores the necessity for caution when dealing with certain function types, notably those featuring plateaus. To address this challenge, a heuristic enhancement is incorporated, improving accuracy in flat regions. This algorithm has been extensively used and tested over the years. NumCosmo includes a comprehensive set of unit tests that rigorously evaluate the algorithm both directly and indirectly, underscoring its robustness and reliability. As a practical application, we compute the surface mass density Sigma(R) and the average surface mass density Sigma(<R) for Navarro-Frenk-White and Hernquist halo density profiles, which provide analytical benchmarks. (abridged)

This paper revisits the problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets. Computing an ECDF at one evaluation point requires O(N) operations on a dataset composed of N data points. Therefore, a direct evaluation of ECDFs at N evaluation points requires a quadratic O(N^2) operations, which is prohibitive for large-scale problems. Two fast and exact methods are proposed and compared. The first one is based on fast summation in lexicographical order, with a O(N{log}N) complexity and requires the evaluation points to lie on a regular grid. The second one is based on the divide-and-conquer principle, with a O(Nlog(N)^{(d-1){vee}1}) complexity and requires the evaluation points to coincide with the input points. The two fast algorithms are described and detailed in the general d-dimensional case, and numerical experiments validate their speed and accuracy. Secondly, the paper establishes a direct connection between cumulative distribution functions and kernel density estimation (KDE) for a large class of kernels. This connection paves the way for fast exact algorithms for multivariate kernel density estimation and kernel regression. Numerical tests with the Laplacian kernel validate the speed and accuracy of the proposed algorithms. A broad range of large-scale multivariate density estimation, cumulative distribution estimation, survival function estimation and regression problems can benefit from the proposed numerical methods.

Open set label shift (OSLS) occurs when label distributions change from a source to a target distribution, and the target distribution has an additional out-of-distribution (OOD) class. In this work, we build estimators for both source and target open set label distributions using a source domain in-distribution (ID) classifier and an ID/OOD classifier. With reasonable assumptions on the ID/OOD classifier, the estimators are assembled into a sequence of three stages: 1) an estimate of the source label distribution of the OOD class, 2) an EM algorithm for Maximum Likelihood estimates (MLE) of the target label distribution, and 3) an estimate of the target label distribution of OOD class under relaxed assumptions on the OOD classifier. The sampling errors of estimates in 1) and 3) are quantified with a concentration inequality. The estimation result allows us to correct the ID classifier trained on the source distribution to the target distribution without retraining. Experiments on a variety of open set label shift settings demonstrate the effectiveness of our model. Our code is available at https://github.com/ChangkunYe/OpenSetLabelShift.

We study a setting of collecting and learning from private data distributed across end users. In the shuffled model of differential privacy, the end users partially protect their data locally before sharing it, and their data is also anonymized during its collection to enhance privacy. This model has recently become a prominent alternative to central DP, which requires full trust in a central data curator, and local DP, where fully local data protection takes a steep toll on downstream accuracy. Our main technical result is a shuffled DP protocol for privately estimating the kernel density function of a distributed dataset, with accuracy essentially matching central DP. We use it to privately learn a classifier from the end user data, by learning a private density function per class. Moreover, we show that the density function itself can recover the semantic content of its class, despite having been learned in the absence of any unprotected data. Our experiments show the favorable downstream performance of our approach, and highlight key downstream considerations and trade-offs in a practical ML deployment of shuffled DP.

We tackle the problem of sampling from intractable high-dimensional density functions, a fundamental task that often appears in machine learning and statistics. We extend recent sampling-based approaches that leverage controlled stochastic processes to model approximate samples from these target densities. The main drawback of these approaches is that the training objective requires full trajectories to compute, resulting in sluggish credit assignment issues due to use of entire trajectories and a learning signal present only at the terminal time. In this work, we present Diffusion Generative Flow Samplers (DGFS), a sampling-based framework where the learning process can be tractably broken down into short partial trajectory segments, via parameterizing an additional "flow function". Our method takes inspiration from the theory developed for generative flow networks (GFlowNets), allowing us to make use of intermediate learning signals. Through various challenging experiments, we demonstrate that DGFS achieves more accurate estimates of the normalization constant than closely-related prior methods.

We present a novel approach for generating minority samples that live on low-density regions of a data manifold. Our framework is built upon diffusion models, leveraging the principle of guided sampling that incorporates an arbitrary energy-based guidance during inference time. The key defining feature of our sampler lies in its self-contained nature, \ie, implementable solely with a pretrained model. This distinguishes our sampler from existing techniques that require expensive additional components (like external classifiers) for minority generation. Specifically, we first estimate the likelihood of features within an intermediate latent sample by evaluating a reconstruction loss w.r.t. its posterior mean. The generation then proceeds with the minimization of the estimated likelihood, thereby encouraging the emergence of minority features in the latent samples of subsequent timesteps. To further improve the performance of our sampler, we provide several time-scheduling techniques that properly manage the influence of guidance over inference steps. Experiments on benchmark real datasets demonstrate that our approach can greatly improve the capability of creating realistic low-likelihood minority instances over the existing techniques without the reliance on costly additional elements. Code is available at https://github.com/soobin-um/sg-minority.

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

Devising indicative evaluation metrics for the image generation task remains an open problem. The most widely used metric for measuring the similarity between real and generated images has been the Fr\'echet Inception Distance (FID) score. Because it does not differentiate the fidelity and diversity aspects of the generated images, recent papers have introduced variants of precision and recall metrics to diagnose those properties separately. In this paper, we show that even the latest version of the precision and recall metrics are not reliable yet. For example, they fail to detect the match between two identical distributions, they are not robust against outliers, and the evaluation hyperparameters are selected arbitrarily. We propose density and coverage metrics that solve the above issues. We analytically and experimentally show that density and coverage provide more interpretable and reliable signals for practitioners than the existing metrics. Code: https://github.com/clovaai/generative-evaluation-prdc.

MDPs with low-rank transitions -- that is, the transition matrix can be factored into the product of two matrices, left and right -- is a highly representative structure that enables tractable learning. The left matrix enables expressive function approximation for value-based learning and has been studied extensively. In this work, we instead investigate sample-efficient learning with density features, i.e., the right matrix, which induce powerful models for state-occupancy distributions. This setting not only sheds light on leveraging unsupervised learning in RL, but also enables plug-in solutions for convex RL. In the offline setting, we propose an algorithm for off-policy estimation of occupancies that can handle non-exploratory data. Using this as a subroutine, we further devise an online algorithm that constructs exploratory data distributions in a level-by-level manner. As a central technical challenge, the additive error of occupancy estimation is incompatible with the multiplicative definition of data coverage. In the absence of strong assumptions like reachability, this incompatibility easily leads to exponential error blow-up, which we overcome via novel technical tools. Our results also readily extend to the representation learning setting, when the density features are unknown and must be learned from an exponentially large candidate set.

The estimation of repeatedly nested expectations is a challenging task that arises in many real-world systems. However, existing methods generally suffer from high computational costs when the number of nestings becomes large. Fix any non-negative integer D for the total number of nestings. Standard Monte Carlo methods typically cost at least O(varepsilon^{-(2+D)}) and sometimes O(varepsilon^{-2(1+D)}) to obtain an estimator up to varepsilon-error. More advanced methods, such as multilevel Monte Carlo, currently only exist for D = 1. In this paper, we propose a novel Monte Carlo estimator called READ, which stands for "Recursive Estimator for Arbitrary Depth.'' Our estimator has an optimal computational cost of O(varepsilon^{-2}) for every fixed D under suitable assumptions, and a nearly optimal computational cost of O(varepsilon^{-2(1 + delta)}) for any 0 < delta < frac12 under much more general assumptions. Our estimator is also unbiased, which makes it easy to parallelize. The key ingredients in our construction are an observation of the problem's recursive structure and the recursive use of the randomized multilevel Monte Carlo method.

In this work, we derive sharp non-asymptotic deviation bounds for weighted sums of Dirichlet random variables. These bounds are based on a novel integral representation of the density of a weighted Dirichlet sum. This representation allows us to obtain a Gaussian-like approximation for the sum distribution using geometry and complex analysis methods. Our results generalize similar bounds for the Beta distribution obtained in the seminal paper Alfers and Dinges [1984]. Additionally, our results can be considered a sharp non-asymptotic version of the inverse of Sanov's theorem studied by Ganesh and O'Connell [1999] in the Bayesian setting. Based on these results, we derive new deviation bounds for the Dirichlet process posterior means with application to Bayesian bootstrap. Finally, we apply our estimates to the analysis of the Multinomial Thompson Sampling (TS) algorithm in multi-armed bandits and significantly sharpen the existing regret bounds by making them independent of the size of the arms distribution support.

Real-world classifiers can benefit from the option of abstaining from predicting on samples where they have low confidence. Such abstention is particularly useful on samples which are close to the learned decision boundary, or which are outliers with respect to the training sample. These settings have been the subject of extensive but disjoint study in the selective classification (SC) and out-of-distribution (OOD) detection literature. Recent work on selective classification with OOD detection (SCOD) has argued for the unified study of these problems; however, the formal underpinnings of this problem are still nascent, and existing techniques are heuristic in nature. In this paper, we propose new plugin estimators for SCOD that are theoretically grounded, effective, and generalise existing approaches from the SC and OOD detection literature. In the course of our analysis, we formally explicate how na\"{i}ve use of existing SC and OOD detection baselines may be inadequate for SCOD. We empirically demonstrate that our approaches yields competitive SC and OOD detection performance compared to baselines from both literatures.

We present an accelerated algorithm for hierarchical density based clustering. Our new algorithm improves upon HDBSCAN*, which itself provided a significant qualitative improvement over the popular DBSCAN algorithm. The accelerated HDBSCAN* algorithm provides comparable performance to DBSCAN, while supporting variable density clusters, and eliminating the need for the difficult to tune distance scale parameter. This makes accelerated HDBSCAN* the default choice for density based clustering. Library available at: https://github.com/scikit-learn-contrib/hdbscan

We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.

Recent advances in diffusion models bring state-of-the-art performance on image generation tasks. However, empirical results from previous research in diffusion models imply an inverse correlation between density estimation and sample generation performances. This paper investigates with sufficient empirical evidence that such inverse correlation happens because density estimation is significantly contributed by small diffusion time, whereas sample generation mainly depends on large diffusion time. However, training a score network well across the entire diffusion time is demanding because the loss scale is significantly imbalanced at each diffusion time. For successful training, therefore, we introduce Soft Truncation, a universally applicable training technique for diffusion models, that softens the fixed and static truncation hyperparameter into a random variable. In experiments, Soft Truncation achieves state-of-the-art performance on CIFAR-10, CelebA, CelebA-HQ 256x256, and STL-10 datasets.

Flow-based generative models are powerful exact likelihood models with efficient sampling and inference. Despite their computational efficiency, flow-based models generally have much worse density modeling performance compared to state-of-the-art autoregressive models. In this paper, we investigate and improve upon three limiting design choices employed by flow-based models in prior work: the use of uniform noise for dequantization, the use of inexpressive affine flows, and the use of purely convolutional conditioning networks in coupling layers. Based on our findings, we propose Flow++, a new flow-based model that is now the state-of-the-art non-autoregressive model for unconditional density estimation on standard image benchmarks. Our work has begun to close the significant performance gap that has so far existed between autoregressive models and flow-based models. Our implementation is available at https://github.com/aravindsrinivas/flowpp

A key challenge of modern machine learning systems is to achieve Out-of-Distribution (OOD) generalization -- generalizing to target data whose distribution differs from that of source data. Despite its significant importance, the fundamental question of ``what are the most effective algorithms for OOD generalization'' remains open even under the standard setting of covariate shift. This paper addresses this fundamental question by proving that, surprisingly, classical Maximum Likelihood Estimation (MLE) purely using source data (without any modification) achieves the minimax optimality for covariate shift under the well-specified setting. That is, no algorithm performs better than MLE in this setting (up to a constant factor), justifying MLE is all you need. Our result holds for a very rich class of parametric models, and does not require any boundedness condition on the density ratio. We illustrate the wide applicability of our framework by instantiating it to three concrete examples -- linear regression, logistic regression, and phase retrieval. This paper further complement the study by proving that, under the misspecified setting, MLE is no longer the optimal choice, whereas Maximum Weighted Likelihood Estimator (MWLE) emerges as minimax optimal in certain scenarios.

Detecting objects in aerial images is challenging because they are typically composed of crowded small objects distributed non-uniformly over high-resolution images. Density cropping is a widely used method to improve this small object detection where the crowded small object regions are extracted and processed in high resolution. However, this is typically accomplished by adding other learnable components, thus complicating the training and inference over a standard detection process. In this paper, we propose an efficient Cascaded Zoom-in (CZ) detector that re-purposes the detector itself for density-guided training and inference. During training, density crops are located, labeled as a new class, and employed to augment the training dataset. During inference, the density crops are first detected along with the base class objects, and then input for a second stage of inference. This approach is easily integrated into any detector, and creates no significant change in the standard detection process, like the uniform cropping approach popular in aerial image detection. Experimental results on the aerial images of the challenging VisDrone and DOTA datasets verify the benefits of the proposed approach. The proposed CZ detector also provides state-of-the-art results over uniform cropping and other density cropping methods on the VisDrone dataset, increasing the detection mAP of small objects by more than 3 points.

We study the problem of training diffusion models to sample from a distribution with a given unnormalized density or energy function. We benchmark several diffusion-structured inference methods, including simulation-based variational approaches and off-policy methods (continuous generative flow networks). Our results shed light on the relative advantages of existing algorithms while bringing into question some claims from past work. We also propose a novel exploration strategy for off-policy methods, based on local search in the target space with the use of a replay buffer, and show that it improves the quality of samples on a variety of target distributions. Our code for the sampling methods and benchmarks studied is made public at https://github.com/GFNOrg/gfn-diffusion as a base for future work on diffusion models for amortized inference.

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In the considered model, a usual likelihood approach can fail to estimate the target distribution consistently due to the singularity. We prove that a novel and effective solution exists by perturbing the data with an instance noise, which leads to consistent estimation of the underlying distribution with desirable convergence rates. We also characterize the class of distributions that can be efficiently estimated via deep generative models. This class is sufficiently general to contain various structured distributions such as product distributions, classically smooth distributions and distributions supported on a low-dimensional manifold. Our analysis provides some insights on how deep generative models can avoid the curse of dimensionality for nonparametric distribution estimation. We conduct a thorough simulation study and real data analysis to empirically demonstrate that the proposed data perturbation technique improves the estimation performance significantly.

Diffusion models are generative models that have recently demonstrated impressive performances in terms of sampling quality and density estimation in high dimensions. They rely on a forward continuous diffusion process and a backward continuous denoising process, which can be described by a time-dependent vector field and is used as a generative model. In the original formulation of the diffusion model, this vector field is assumed to be the score function (i.e. it is the gradient of the log-probability at a given time in the diffusion process). Curiously, on the practical side, most studies on diffusion models implement this vector field as a neural network function and do not constrain it be the gradient of some energy function (that is, most studies do not constrain the vector field to be conservative). Even though some studies investigated empirically whether such a constraint will lead to a performance gain, they lead to contradicting results and failed to provide analytical results. Here, we provide three analytical results regarding the extent of the modeling freedom of this vector field. {Firstly, we propose a novel decomposition of vector fields into a conservative component and an orthogonal component which satisfies a given (gauge) freedom. Secondly, from this orthogonal decomposition, we show that exact density estimation and exact sampling is achieved when the conservative component is exactly equals to the true score and therefore conservativity is neither necessary nor sufficient to obtain exact density estimation and exact sampling. Finally, we show that when it comes to inferring local information of the data manifold, constraining the vector field to be conservative is desirable.

Object counting methods typically rely on manually annotated datasets. The cost of creating such datasets has restricted the versatility of these networks to count objects from specific classes (such as humans or penguins), and counting objects from diverse categories remains a challenge. The availability of robust text-to-image latent diffusion models (LDMs) raises the question of whether these models can be utilized to generate counting datasets. However, LDMs struggle to create images with an exact number of objects based solely on text prompts but they can be used to offer a dependable sorting signal by adding and removing objects within an image. Leveraging this data, we initially introduce an unsupervised sorting methodology to learn object-related features that are subsequently refined and anchored for counting purposes using counting data generated by LDMs. Further, we present a density classifier-guided method for dividing an image into patches containing objects that can be reliably counted. Consequently, we can generate counting data for any type of object and count them in an unsupervised manner. Our approach outperforms other unsupervised and few-shot alternatives and is not restricted to specific object classes for which counting data is available. Code to be released upon acceptance.

Estimating mutual information (MI) is a fundamental task in data science and machine learning. Existing estimators mainly rely on either highly flexible models (e.g., neural networks), which require large amounts of data, or overly simplified models (e.g., Gaussian copula), which fail to capture complex distributions. Drawing upon recent vector copula theory, we propose a principled interpolation between these two extremes to achieve a better trade-off between complexity and capacity. Experiments on state-of-the-art synthetic benchmarks and real-world data with diverse modalities demonstrate the advantages of the proposed estimator.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

We explore a generative machine learning-based approach for estimating multi-dimensional probability density functions (PDFs) in a target sample using a statistically independent but related control sample - a common challenge in particle physics data analysis. The generative model must accurately reproduce individual observable distributions while preserving the correlations between them, based on the input multidimensional distribution from the control sample. Here we present a conditional normalizing flow model (CNF) based on a chain of bijectors which learns to transform unpaired simulation events to data events. We assess the performance of the CNF model in the context of LHC Higgs to diphoton analysis, where we use the CNF model to convert a Monte Carlo diphoton sample to one that models data. We show that the CNF model can accurately model complex data distributions and correlations. We also leverage the recently popularized Modified Differential Multiplier Method (MDMM) to improve the convergence of our model and assign physical meaning to usually arbitrary loss-function parameters.

A great deal of effort has been devoted to reducing the risk of spurious scientific discoveries, from the use of sophisticated validation techniques, to deep statistical methods for controlling the false discovery rate in multiple hypothesis testing. However, there is a fundamental disconnect between the theoretical results and the practice of data analysis: the theory of statistical inference assumes a fixed collection of hypotheses to be tested, or learning algorithms to be applied, selected non-adaptively before the data are gathered, whereas in practice data is shared and reused with hypotheses and new analyses being generated on the basis of data exploration and the outcomes of previous analyses. In this work we initiate a principled study of how to guarantee the validity of statistical inference in adaptive data analysis. As an instance of this problem, we propose and investigate the question of estimating the expectations of m adaptively chosen functions on an unknown distribution given n random samples. We show that, surprisingly, there is a way to estimate an exponential in n number of expectations accurately even if the functions are chosen adaptively. This gives an exponential improvement over standard empirical estimators that are limited to a linear number of estimates. Our result follows from a general technique that counter-intuitively involves actively perturbing and coordinating the estimates, using techniques developed for privacy preservation. We give additional applications of this technique to our question.

Active learning is a machine learning paradigm that aims to improve the performance of a model by strategically selecting and querying unlabeled data. One effective selection strategy is to base it on the model's predictive uncertainty, which can be interpreted as a measure of how informative a sample is. The sample's distance to the decision boundary is a natural measure of predictive uncertainty, but it is often intractable to compute, especially for complex decision boundaries formed in multiclass classification tasks. To address this issue, this paper proposes the {\it least disagree metric} (LDM), defined as the smallest probability of disagreement of the predicted label, and an estimator for LDM proven to be asymptotically consistent under mild assumptions. The estimator is computationally efficient and can be easily implemented for deep learning models using parameter perturbation. The LDM-based active learning is performed by querying unlabeled data with the smallest LDM. Experimental results show that our LDM-based active learning algorithm obtains state-of-the-art overall performance on all considered datasets and deep architectures.

Finding the mode of a high dimensional probability distribution D is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when D is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.'s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy (1-epsilon) for any epsilon > 0. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless P = NP. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction.

This work derives methods for performing nonparametric, nonasymptotic statistical inference for population means under the constraint of local differential privacy (LDP). Given bounded observations (X_1, dots, X_n) with mean mu^star that are privatized into (Z_1, dots, Z_n), we present confidence intervals (CI) and time-uniform confidence sequences (CS) for mu^star when only given access to the privatized data. To achieve this, we introduce a nonparametric and sequentially interactive generalization of Warner's famous ``randomized response'' mechanism, satisfying LDP for arbitrary bounded random variables, and then provide CIs and CSs for their means given access to the resulting privatized observations. For example, our results yield private analogues of Hoeffding's inequality in both fixed-time and time-uniform regimes. We extend these Hoeffding-type CSs to capture time-varying (non-stationary) means, and conclude by illustrating how these methods can be used to conduct private online A/B tests.

Joint Embedding Predictive Architectures (JEPAs) learn representations able to solve numerous downstream tasks out-of-the-box. JEPAs combine two objectives: (i) a latent-space prediction term, i.e., the representation of a slightly perturbed sample must be predictable from the original sample's representation, and (ii) an anti-collapse term, i.e., not all samples should have the same representation. While (ii) is often considered as an obvious remedy to representation collapse, we uncover that JEPAs' anti-collapse term does much more--it provably estimates the data density. In short, any successfully trained JEPA can be used to get sample probabilities, e.g., for data curation, outlier detection, or simply for density estimation. Our theoretical finding is agnostic of the dataset and architecture used--in any case one can compute the learned probabilities of sample x efficiently and in closed-form using the model's Jacobian matrix at x. Our findings are empirically validated across datasets (synthetic, controlled, and Imagenet) and across different Self Supervised Learning methods falling under the JEPA family (I-JEPA and DINOv2) and on multimodal models, such as MetaCLIP. We denote the method extracting the JEPA learned density as {\bf JEPA-SCORE}.

Multi-agent systems play a central role in area coverage tasks across search-and-rescue, environmental monitoring, and precision agriculture. Achieving non-uniform coverage, where spatial priorities vary across the domain, requires coordinating agents while respecting dynamic and communication constraints. Density-driven approaches can distribute agents according to a prescribed reference density, but existing methods do not ensure connectivity. This limitation often leads to communication loss, reduced coordination, and degraded coverage performance. This letter introduces a connectivity-preserving extension of the Density-Driven Optimal Control (D2OC) framework. The coverage objective, defined using the Wasserstein distance between the agent distribution and the reference density, admits a convex quadratic program formulation. Communication constraints are incorporated through a smooth connectivity penalty, which maintains strict convexity, supports distributed implementation, and preserves inter-agent communication without imposing rigid formations. Simulation studies show that the proposed method consistently maintains connectivity, improves convergence speed, and enhances non-uniform coverage quality compared with density-driven schemes that do not incorporate explicit connectivity considerations.

The standard Monte Carlo estimator I_N^{MC} of int fdω relies on independent samples from ω and has variance of order 1/N. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to f and ω. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of O(N^{-(1+1/d)}) for smooth f, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order 1/N, like Monte Carlo, but its DPP is tailored to f. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.

Inference-time steering enables pretrained diffusion/flow models to be adapted to new tasks without retraining. A widely used approach is the ratio-of-densities method, which defines a time-indexed target path by reweighting probability-density trajectories from multiple models with positive, or in some cases, negative exponents. This construction, however, harbors a critical and previously unformalized failure mode: Marginal Path Collapse, where intermediate densities become non-normalizable even though endpoints remain valid. Collapse arises systematically when composing heterogeneous models trained on different noise schedules or datasets, including a common setting in molecular design where de-novo, conformer, and pocket-conditioned models must be combined for tasks such as flexible-pose scaffold decoration. We provide a novel and complete solution for the problem. First, we derive a simple path existence criterion that predicts exactly when collapse occurs from noise schedules and exponents alone. Second, we introduce Adaptive path Correction with Exponents (ACE), which extends Feynman-Kac steering to time-varying exponents and guarantees a valid probability path. On a synthetic 2D benchmark and on flexible-pose scaffold decoration, ACE eliminates collapse and enables high-guidance compositional generation, improving distributional and docking metrics over constant-exponent baselines and even specialized task-specific scaffold decoration models. Our work turns ratio-of-densities steering with heterogeneous experts from an unstable heuristic into a reliable tool for controllable generation.

Classical state estimation algorithms rely on predefined target's state-space model, which complicates model derivation and limits adaptability when system dynamics change. Neural network based estimators offer a data-driven alternative, but rarely fuse classical estimation theory into their structure and demand large, pre-computed training sets. To overcome these limitations, we propose a unified state-space structure without target's state-space model and treats both the input-layer activations and all network weights as latent states to be estimated online. We instantiate this nonlinear model with three canonical estimators-the extended Kalman estimator, the unscented Kalman estimator, and the particle estimator to simulate different neural network and demonstrate its generality. We then benchmark our approach against seven leading neural network estimators across three representative scenarios. Results show that our neural network state-space estimators not only retain the robust learning capability, but also match or exceed the accuracy of both classical and pre-trained neural network methods. Code, data, and more result: github.com/ShineMinxing/PaperNNSSE.git

Solving complex classification tasks using deep neural networks typically requires large amounts of annotated data. However, corresponding class labels are noisy when provided by error-prone annotators, e.g., crowd workers. Training standard deep neural networks leads to subpar performances in such multi-annotator supervised learning settings. We address this issue by presenting a probabilistic training framework named multi-annotator deep learning (MaDL). A ground truth and an annotator performance model are jointly trained in an end-to-end learning approach. The ground truth model learns to predict instances' true class labels, while the annotator performance model infers probabilistic estimates of annotators' performances. A modular network architecture enables us to make varying assumptions regarding annotators' performances, e.g., an optional class or instance dependency. Further, we learn annotator embeddings to estimate annotators' densities within a latent space as proxies of their potentially correlated annotations. Together with a weighted loss function, we improve the learning from correlated annotation patterns. In a comprehensive evaluation, we examine three research questions about multi-annotator supervised learning. Our findings indicate MaDL's state-of-the-art performance and robustness against many correlated, spamming annotators.

Stochastic neurons and hard non-linearities can be useful for a number of reasons in deep learning models, but in many cases they pose a challenging problem: how to estimate the gradient of a loss function with respect to the input of such stochastic or non-smooth neurons? I.e., can we "back-propagate" through these stochastic neurons? We examine this question, existing approaches, and compare four families of solutions, applicable in different settings. One of them is the minimum variance unbiased gradient estimator for stochatic binary neurons (a special case of the REINFORCE algorithm). A second approach, introduced here, decomposes the operation of a binary stochastic neuron into a stochastic binary part and a smooth differentiable part, which approximates the expected effect of the pure stochatic binary neuron to first order. A third approach involves the injection of additive or multiplicative noise in a computational graph that is otherwise differentiable. A fourth approach heuristically copies the gradient with respect to the stochastic output directly as an estimator of the gradient with respect to the sigmoid argument (we call this the straight-through estimator). To explore a context where these estimators are useful, we consider a small-scale version of {\em conditional computation}, where sparse stochastic units form a distributed representation of gaters that can turn off in combinatorially many ways large chunks of the computation performed in the rest of the neural network. In this case, it is important that the gating units produce an actual 0 most of the time. The resulting sparsity can be potentially be exploited to greatly reduce the computational cost of large deep networks for which conditional computation would be useful.

In location estimation, we are given n samples from a known distribution f shifted by an unknown translation lambda, and want to estimate lambda as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cram\'er-Rao bound of error mathcal N(0, 1{nmathcal I}), where mathcal I is the Fisher information of f. However, the n required for convergence depends on f, and may be arbitrarily large. We build on the theory using smoothed estimators to bound the error for finite n in terms of mathcal I_r, the Fisher information of the r-smoothed distribution. As n to infty, r to 0 at an explicit rate and this converges to the Cram\'er-Rao bound. We (1) improve the prior work for 1-dimensional f to converge for constant failure probability in addition to high probability, and (2) extend the theory to high-dimensional distributions. In the process, we prove a new bound on the norm of a high-dimensional random variable whose 1-dimensional projections are subgamma, which may be of independent interest.

The paper introduces the weighted convolution, a novel approach to the convolution for signals defined on regular grids (e.g., 2D images) through the application of an optimal density function to scale the contribution of neighbouring pixels based on their distance from the central pixel. This choice differs from the traditional uniform convolution, which treats all neighbouring pixels equally. Our weighted convolution can be applied to convolutional neural network problems to improve the approximation accuracy. Given a convolutional network, we define a framework to compute the optimal density function through a minimisation model. The framework separates the optimisation of the convolutional kernel weights (using stochastic gradient descent) from the optimisation of the density function (using DIRECT-L). Experimental results on a learning model for an image-to-image task (e.g., image denoising) show that the weighted convolution significantly reduces the loss (up to 53% improvement) and increases the test accuracy compared to standard convolution. While this method increases execution time by 11%, it is robust across several hyperparameters of the learning model. Future work will apply the weighted convolution to real-case 2D and 3D image convolutional learning problems.

Diffusion-based generative models (DBGMs) perturb data to a target noise distribution and reverse this process to generate samples. The choice of noising process, or inference diffusion process, affects both likelihoods and sample quality. For example, extending the inference process with auxiliary variables leads to improved sample quality. While there are many such multivariate diffusions to explore, each new one requires significant model-specific analysis, hindering rapid prototyping and evaluation. In this work, we study Multivariate Diffusion Models (MDMs). For any number of auxiliary variables, we provide a recipe for maximizing a lower-bound on the MDMs likelihood without requiring any model-specific analysis. We then demonstrate how to parameterize the diffusion for a specified target noise distribution; these two points together enable optimizing the inference diffusion process. Optimizing the diffusion expands easy experimentation from just a few well-known processes to an automatic search over all linear diffusions. To demonstrate these ideas, we introduce two new specific diffusions as well as learn a diffusion process on the MNIST, CIFAR10, and ImageNet32 datasets. We show learned MDMs match or surpass bits-per-dims (BPDs) relative to fixed choices of diffusions for a given dataset and model architecture.

The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) -- an arbitrary sequence of deterministic invertible functions and stochastic sampling blocks. We show that stochasticity overcomes expressivity limitations of normalizing flows resulting from the invertibility constraint, whereas trainable transformations between sampling steps improve efficiency of pure MCMC/LD along the flow. By invoking ideas from non-equilibrium statistical mechanics we derive an efficient training procedure by which both the sampler's and the flow's parameters can be optimized end-to-end, and by which we can compute exact importance weights without having to marginalize out the randomness of the stochastic blocks. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks including applications to sampling molecular systems in equilibrium.

Parameter estimation for gravitational-wave signals is computationally demanding due to the high dimensionality of the parameter space and the cost of repeated waveform generation in traditional Bayesian inference. These analyses require on the order of 10^8 likelihood evaluations and waveform generations, resulting in inference times of hours to days per event. Furthermore, discrepancies between waveform models introduce systematic uncertainties that can bias inferred source properties. To address these challenges, we propose a novel framework based on Simulation-based Inference (SBI) and Neural Posterior Estimation (NPE) and apply it to signals from Intermediate-Mass Black Holes (IMBH). In this framework, we train a single amortised neural posterior estimator on a large simulated dataset generated using two state-of-the-art waveform approximants, IMRPhenomXPHM and SEOBNRv5PHM. By treating the waveform model index as a latent variable, the network learns to produce posterior distributions that are naturally marginalized over the discrepancies of the two waveform models. Once trained, the model enables direct posterior sampling in milliseconds per event, eliminating the need for likelihood evaluations while simultaneously accounting for model systematics. We demonstrate that this approach recovers accurate posterior distributions for IMBH signals injected into Gaussian noise, achieving close agreement with traditional nested-sampling results while reducing inference time by several orders of magnitude. Our results show that NPE can robustly incorporate waveform-model systematics within a unified framework, offering a scalable path toward rapid, systematics-aware gravitational-wave inference. Establishing these methods as promising alternatives to classical likelihood-based pipelines for current and future high-mass gravitational-wave observations.

The training of large language models (LLMs) is expensive. In this paper, we study data-efficient approaches for pre-training LLMs, i.e., techniques that aim to optimize the Pareto frontier of model quality and training resource/data consumption. We seek to understand the tradeoffs associated with data selection routines based on (i) expensive-to-compute data-quality estimates, and (ii) maximization of coverage and diversity-based measures in the feature space. Our first technique, Ask-LLM, leverages the zero-shot reasoning capabilities of instruction-tuned LLMs to directly assess the quality of a training example. To target coverage, we propose Density sampling, which models the data distribution to select a diverse sample. In our comparison of 19 samplers, involving hundreds of evaluation tasks and pre-training runs, we find that Ask-LLM and Density are the best methods in their respective categories. Coverage sampling can recover the performance of the full data, while models trained on Ask-LLM data consistently outperform full-data training -- even when we reject 90% of the original dataset, while converging up to 70% faster.

The SIML (abbreviation of Separating Information Maximal Likelihood) method, has been introduced by N. Kunitomo and S. Sato and their collaborators to estimate the integrated volatility of high-frequency data that is assumed to be an It\^o process but with so-called microstructure noise. The SIML estimator turned out to share many properties with the estimator introduced by P. Malliavin and M.E. Mancino. The present paper establishes the consistency and the asymptotic normality under a general sampling scheme but without microstructure noise. Specifically, a fast convergence shown for Malliavin--Mancino estimator by E. Clement and A. Gloter is also established for the SIML estimator.

In this paper, we present a novel approach to accelerate the Bayesian inference process, focusing specifically on the nested sampling algorithms. Bayesian inference plays a crucial role in cosmological parameter estimation, providing a robust framework for extracting theoretical insights from observational data. However, its computational demands can be substantial, primarily due to the need for numerous likelihood function evaluations. Our proposed method utilizes the power of deep learning, employing feedforward neural networks to approximate the likelihood function dynamically during the Bayesian inference process. Unlike traditional approaches, our method trains neural networks on-the-fly using the current set of live points as training data, without the need for pre-training. This flexibility enables adaptation to various theoretical models and datasets. We perform simple hyperparameter optimization using genetic algorithms to suggest initial neural network architectures for learning each likelihood function. Once sufficient accuracy is achieved, the neural network replaces the original likelihood function. The implementation integrates with nested sampling algorithms and has been thoroughly evaluated using both simple cosmological dark energy models and diverse observational datasets. Additionally, we explore the potential of genetic algorithms for generating initial live points within nested sampling inference, opening up new avenues for enhancing the efficiency and effectiveness of Bayesian inference methods.

Localizing individuals in crowds is more in accordance with the practical demands of subsequent high-level crowd analysis tasks than simply counting. However, existing localization based methods relying on intermediate representations (i.e., density maps or pseudo boxes) serving as learning targets are counter-intuitive and error-prone. In this paper, we propose a purely point-based framework for joint crowd counting and individual localization. For this framework, instead of merely reporting the absolute counting error at image level, we propose a new metric, called density Normalized Average Precision (nAP), to provide more comprehensive and more precise performance evaluation. Moreover, we design an intuitive solution under this framework, which is called Point to Point Network (P2PNet). P2PNet discards superfluous steps and directly predicts a set of point proposals to represent heads in an image, being consistent with the human annotation results. By thorough analysis, we reveal the key step towards implementing such a novel idea is to assign optimal learning targets for these proposals. Therefore, we propose to conduct this crucial association in an one-to-one matching manner using the Hungarian algorithm. The P2PNet not only significantly surpasses state-of-the-art methods on popular counting benchmarks, but also achieves promising localization accuracy. The codes will be available at: https://github.com/TencentYoutuResearch/CrowdCounting-P2PNet.

The study conducted by Shumailov et al. (2024) demonstrates that repeatedly training a generative model on synthetic data leads to model collapse. This finding has generated considerable interest and debate, particularly given that current models have nearly exhausted the available data. In this work, we investigate the effects of fitting a distribution (through Kernel Density Estimation, or KDE) or a model to the data, followed by repeated sampling from it. Our objective is to develop a theoretical understanding of the phenomenon observed by Shumailov et al. (2024). Our results indicate that the outcomes reported are a statistical phenomenon and may be unavoidable.

Causal effects are usually studied in terms of the means of counterfactual distributions, which may be insufficient in many scenarios. Given a class of densities known up to normalizing constants, we propose to model counterfactual distributions by minimizing kernel Stein discrepancies in a doubly robust manner. This enables the estimation of counterfactuals over large classes of distributions while exploiting the desired double robustness. We present a theoretical analysis of the proposed estimator, providing sufficient conditions for consistency and asymptotic normality, as well as an examination of its empirical performance.

Discrete diffusion models offer global context awareness and flexible parallel generation. However, uniform random noise schedulers in standard DLLM training overlook the highly non-uniform information density inherent in real-world sequences. This wastes optimization resources on low-density structural glues while leaving high-density logical pivot points severely under-optimized. To address this, we propose an Information Density Driven Smart Noise Scheduler. By extracting information-dense hubs and applying Complementary Priority Masking, our method decouples a single training instance into mutually reinforcing reasoning and syntax samples, forcing the model to master both logical deduction and foundational sequence structure. Experiments demonstrate that our approach improves average accuracy by ~4\% across four Code and Math reasoning benchmarks, significantly outperforming uniform baselines. Mechanistic analyses further reveal that probabilistic priority masking effectively mitigates contextual collapse during block diffusion training. Overall, this density-aware strategy efficiently unlocks the reasoning potential of diffusion language models at minimal annotation cost, emerging as a promising new masked data training paradigm for Diffusion LLMs. Our processed dataset can be found at https://huggingface.co/datasets/malr07/opc-sft-stage2-dense-extracted.

Abstractive summarization models are commonly trained using maximum likelihood estimation, which assumes a deterministic (one-point) target distribution in which an ideal model will assign all the probability mass to the reference summary. This assumption may lead to performance degradation during inference, where the model needs to compare several system-generated (candidate) summaries that have deviated from the reference summary. To address this problem, we propose a novel training paradigm which assumes a non-deterministic distribution so that different candidate summaries are assigned probability mass according to their quality. Our method achieves a new state-of-the-art result on the CNN/DailyMail (47.78 ROUGE-1) and XSum (49.07 ROUGE-1) datasets. Further analysis also shows that our model can estimate probabilities of candidate summaries that are more correlated with their level of quality.

Modern ML applications increasingly rely on complex deep learning models and large datasets. There has been an exponential growth in the amount of computation needed to train the largest models. Therefore, to scale computation and data, these models are inevitably trained in a distributed manner in clusters of nodes, and their updates are aggregated before being applied to the model. However, a distributed setup is prone to Byzantine failures of individual nodes, components, and software. With data augmentation added to these settings, there is a critical need for robust and efficient aggregation systems. We define the quality of workers as reconstruction ratios in (0,1], and formulate aggregation as a Maximum Likelihood Estimation procedure using Beta densities. We show that the Regularized form of log-likelihood wrt subspace can be approximately solved using iterative least squares solver, and provide convergence guarantees using recent Convex Optimization landscape results. Our empirical findings demonstrate that our approach significantly enhances the robustness of state-of-the-art Byzantine resilient aggregators. We evaluate our method in a distributed setup with a parameter server, and show simultaneous improvements in communication efficiency and accuracy across various tasks. The code is publicly available at https://github.com/hamidralmasi/FlagAggregator

Distribution inference, sometimes called property inference, infers statistical properties about a training set from access to a model trained on that data. Distribution inference attacks can pose serious risks when models are trained on private data, but are difficult to distinguish from the intrinsic purpose of statistical machine learning -- namely, to produce models that capture statistical properties about a distribution. Motivated by Yeom et al.'s membership inference framework, we propose a formal definition of distribution inference attacks that is general enough to describe a broad class of attacks distinguishing between possible training distributions. We show how our definition captures previous ratio-based property inference attacks as well as new kinds of attack including revealing the average node degree or clustering coefficient of a training graph. To understand distribution inference risks, we introduce a metric that quantifies observed leakage by relating it to the leakage that would occur if samples from the training distribution were provided directly to the adversary. We report on a series of experiments across a range of different distributions using both novel black-box attacks and improved versions of the state-of-the-art white-box attacks. Our results show that inexpensive attacks are often as effective as expensive meta-classifier attacks, and that there are surprising asymmetries in the effectiveness of attacks. Code is available at https://github.com/iamgroot42/FormEstDistRisks

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

The goal of the change-point detection is to discover changes of time series distribution. One of the state of the art approaches of the change-point detection are based on direct density ratio estimation. In this work we show how existing algorithms can be generalized using various binary classification and regression models. In particular, we show that the Gradient Boosting over Decision Trees and Neural Networks can be used for this purpose. The algorithms are tested on several synthetic and real-world datasets. The results show that the proposed methods outperform classical RuLSIF algorithm. Discussion of cases where the proposed algorithms have advantages over existing methods are also provided.

In this paper, we introduce a conformal prediction method to construct prediction sets in a oneshot federated learning setting. More specifically, we define a quantile-of-quantiles estimator and prove that for any distribution, it is possible to output prediction sets with desired coverage in only one round of communication. To mitigate privacy issues, we also describe a locally differentially private version of our estimator. Finally, over a wide range of experiments, we show that our method returns prediction sets with coverage and length very similar to those obtained in a centralized setting. Overall, these results demonstrate that our method is particularly well-suited to perform conformal predictions in a one-shot federated learning setting.

There has been a lot of recent interest in designing neural network models to estimate a distribution from a set of examples. We introduce a simple modification for autoencoder neural networks that yields powerful generative models. Our method masks the autoencoder's parameters to respect autoregressive constraints: each input is reconstructed only from previous inputs in a given ordering. Constrained this way, the autoencoder outputs can be interpreted as a set of conditional probabilities, and their product, the full joint probability. We can also train a single network that can decompose the joint probability in multiple different orderings. Our simple framework can be applied to multiple architectures, including deep ones. Vectorized implementations, such as on GPUs, are simple and fast. Experiments demonstrate that this approach is competitive with state-of-the-art tractable distribution estimators. At test time, the method is significantly faster and scales better than other autoregressive estimators.

We introduce a novel weighted convolution operator that enhances traditional convolutional neural networks (CNNs) by integrating a spatial density function into the convolution operator. This extension enables the network to differentially weight neighbouring pixels based on their relative position to the reference pixel, improving spatial characterisation and feature extraction. The proposed operator maintains the same number of trainable parameters and is fully compatible with existing CNN architectures. Although developed for 2D image data, the framework is generalisable to signals on regular grids of arbitrary dimensions, such as 3D volumetric data or 1D time series. We propose an efficient implementation of the weighted convolution by pre-computing the density function and achieving execution times comparable to standard convolution layers. We evaluate our method on two deep learning tasks: image classification using the CIFAR-100 dataset [KH+09] and image denoising using the DIV2K dataset [AT17]. Experimental results with state-of-the-art classification (e.g., VGG [SZ15], ResNet [HZRS16]) and denoising (e.g., DnCNN [ZZC+17], NAFNet [CCZS22]) methods show that the weighted convolution improves performance with respect to standard convolution across different quantitative metrics. For example, VGG achieves an accuracy of 66.94% with weighted convolution versus 56.89% with standard convolution on the classification problem, while DnCNN improves the PSNR value from 20.17 to 22.63 on the denoising problem. All models were trained on the CINECA Leonardo cluster to reduce the execution time and improve the tuning of the density function values. The PyTorch implementation of the weighted convolution is publicly available at: https://github.com/cammarasana123/weightedConvolution2.0.

Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies to compute forces and higher-order derivatives. Furthermore, such densities are often defined on non-trivial topologies. A recent example are Boltzmann Generators for generating 3D-structures of peptides and small proteins. These generative models leverage the space of internal coordinates (dihedrals, angles, and bonds), which is a product of hypertori and compact intervals. In this work, we introduce a class of smooth mixture transformations working on both compact intervals and hypertori. Mixture transformations employ root-finding methods to invert them in practice, which has so far prevented bi-directional flow training. To this end, we show that parameter gradients and forces of such inverses can be computed from forward evaluations via the inverse function theorem. We demonstrate two advantages of such smooth flows: they allow training by force matching to simulation data and can be used as potentials in molecular dynamics simulations.

In modern machine learning problems we deal with datasets that are either distributed by nature or potentially large for which distributing the computations is usually a standard way to proceed, since centralized algorithms are in general ineffective. We propose a distributed learning approach for mixtures of experts (MoE) models with an aggregation strategy to construct a reduction estimator from local estimators fitted parallelly to distributed subsets of the data. The aggregation is based on an optimal minimization of an expected transportation divergence between the large MoE composed of local estimators and the unknown desired MoE model. We show that the provided reduction estimator is consistent as soon as the local estimators to be aggregated are consistent, and its construction is performed by a proposed majorization-minimization (MM) algorithm that is computationally effective. We study the statistical and numerical properties for the proposed reduction estimator on experiments that demonstrate its performance compared to namely the global estimator constructed in a centralized way from the full dataset. For some situations, the computation time is more than ten times faster, for a comparable performance. Our source codes are publicly available on Github.