Addressing real-world optimization problems becomes particularly challenging when analytic objective functions or constraints are unavailable. While numerous studies have addressed the issue of unknown objectives, limited research has focused on scenarios where feasibility constraints are not given explicitly. Overlooking these constraints can lead to spurious solutions that are unrealistic in practice. To deal with such unknown constraints, we propose to perform optimization within the data manifold using diffusion models. To constrain the optimization process to the data manifold, we reformulate the original optimization problem as a sampling problem from the product of the Boltzmann distribution defined by the objective function and the data distribution learned by the diffusion model. Depending on the differentiability of the objective function, we propose two different sampling methods. For differentiable objectives, we propose a two-stage framework that begins with a guided diffusion process for warm-up, followed by a Langevin dynamics stage for further correction. For non-differentiable objectives, we propose an iterative importance sampling strategy using the diffusion model as the proposal distribution. Comprehensive experiments on a synthetic dataset, six real-world black-box optimization datasets, and a multi-objective molecule optimization dataset show that our method achieves better or comparable performance with previous state-of-the-art baselines.

Sampling and automatic differentiation are both ubiquitous in modern machine learning. At its intersection, differentiating through a sampling operation, with respect to the parameters of the sampling process, is a problem that is both challenging and broadly applicable. We introduce a general framework and a new algorithm for first-order optimization of parameterized stochastic diffusions, performing jointly, in a single loop, optimization and sampling steps. This approach is inspired by recent advances in bilevel optimization and automatic implicit differentiation, leveraging the point of view of sampling as optimization over the space of probability distributions. We provide theoretical and experimental results showcasing the performance of our method.

We propose a gradient flow procedure for generative modeling by transporting particles from an initial source distribution to a target distribution, where the gradient field on the particles is given by a noise-adaptive Wasserstein Gradient of the Maximum Mean Discrepancy (MMD). The noise adaptive MMD is trained on data distributions corrupted by increasing levels of noise, obtained via a forward diffusion process, as commonly used in denoising diffusion probabilistic models. The result is a generalization of MMD Gradient Flow, which we call Diffusion-MMD-Gradient Flow or DMMD. The divergence training procedure is related to discriminator training in Generative Adversarial Networks (GAN), but does not require adversarial training. We obtain competitive empirical performance in unconditional image generation on CIFAR10, MNIST, CELEB-A (64 x64) and LSUN Church (64 x 64). Furthermore, we demonstrate the validity of the approach when MMD is replaced by a lower bound on the KL divergence.

Speculative sampling is a popular technique for accelerating inference in Large Language Models by generating candidate tokens using a fast draft model and then accepting or rejecting them based on the target model’s distribution. While speculative sampling was previously limited to discrete sequences, we extend it to diffusion models, which generate samples via continuous, vector-valued Markov chains. In this context, the target model is a high-quality but computationally expensive diffusion model. We propose various drafting strategies, including a simple and effective approach that does not require training a draft model and is applicable out-of-the-box to any diffusion model. We demonstrate significant generation speedup on various diffusion models, halving the number of function evaluations while generating exact samples from the target model. Finally, we also show how this procedure can be used to accelerate Langevin diffusions to sample unnormalized distributions.

Diffusion models generate high-quality synthetic data. They operate by defining a continuous-time forward process which gradually adds Gaussian noise to data until fully corrupted. The corresponding reverse process progressively “denoises" a Gaussian sample into a sample from the data distribution. However, generating high-quality outputs requires many discretization steps to obtain a faithful approximation of the reverse process. This is expensive and has motivated the development of many acceleration methods. We propose to speed up sample generation by learning the posterior distribution of clean data samples given their noisy versions, instead of only the mean of this distribution. This allows us to sample from the probability transitions of the reverse process on a coarse time scale, significantly accelerating inference with minimal degradation of the quality of the output. This is accomplished by replacing the standard regression loss used to estimate conditional means with a scoring rule. We validate our method on image and robot trajectory generation, where we consistently outperform standard diffusion models at few discretization steps.

Sampling efficiently from a target unnormalized probability density remains a core challenge, with relevance across countless high-impact scientific applications. A promising approach towards this challenge is the design of amortized samplers that borrow key ideas, such as probability path design, from state-of-the-art generative diffusion models. However, all existing diffusion-based samplers remain unable to draw samples from distributions at the scale of even simple molecular systems. In this paper, we propose Progressive Inference-Time Annealing (PITA) a novel framework to learn diffusion-based samplers that combines two complementary interpolation techniques: I.) Annealing of the Boltzmann distribution and II.) Diffusion smoothing. PITA trains a sequence of diffusion models from high to low temperatures by sequentially training each model at progressively higher temperatures, leveraging engineered easy access to samples of the temperature-annealed target density. In the subsequent step, PITA enables simulating the trained diffusion model to *procure training samples at a lower temperature* for the next diffusion model through inference-time annealing using a novel Feynman-Kac PDE combined with Sequential Monte Carlo. Empirically, PITA enables, for the first time, equilibrium sampling of $N$-body particle systems, Alanine Dipeptide, and tripeptides in Cartesian coordinates with dramatically lower energy function evaluations.

When do diffusion models reproduce their training data, and when are they able to generate samples beyond it? A practically relevant theoretical understanding of this interplay between memorization and generalization may significantly impact real-world deployments of diffusion models with respect to issues such as copyright infringement and data privacy. In this paper, we propose a scientific and mathematical “laboratory” for investigating memorization and generalization in diffusion models trained on fully synthetic or natural image-like structured data. Within this setting, we theoretically characterize a crossover point wherein the weighted training loss of a fully generalizing model becomes greater than that of an underparameterized memorizing model at a critical value of model (under)parameterization. We then demonstrate via carefully-designed experiments that the location of this crossover predicts a phase transition in diffusion models trained via gradient descent, as our theory enables us to analytically predict the model size at which memorization becomes predominant. Our work provides an analytically tractable and practically meaningful setting for future theoretical and empirical investigations. Code for our experiments is available at https://github.com/DruvPai/diffusion_mem_gen.

We consider the problem of sampling distributions stemming from non-convex potentials with Unadjusted Langevin Algorithm (ULA). We prove the stability of the discrete-time ULA to drift approximations under the assumption that the potential is strongly convex at infinity. In many context, e.g. imaging inverse problems, potentials are non-convex and non-smooth. Proximal Stochastic Gradient Langevin Algorithm (PSGLA) is a popular algorithm to handle such potentials. It combines the forward-backward optimization algorithm with a ULA step. Our main stability result combined with properties of the Moreau envelope allows us to derive the first proof of convergence of the PSGLA for non-convex potentials. We empirically validate our methodology on synthetic data and in the context of imaging inverse problems. In particular, we observe that PSGLA exhibits faster convergence rates than Stochastic Gradient Langevin Algorithm for posterior sampling while preserving its restoration properties.

Denoising diffusions are a powerful method to generate approximate samples from high-dimensional data distributions. Recent results provide polynomial bounds on their convergence rate, assuming $L^2$-accurate scores. Until now, the tightest bounds were either superlinear in the data dimension or required strong smoothness assumptions. We provide the first convergence bounds which are linear in the data dimension (up to logarithmic factors) assuming only finite second moments of the data distribution. We show that diffusion models require at most $\tilde O(\frac{d \log^2(1/\delta)}{\varepsilon^2})$ steps to approximate an arbitrary distribution on $\mathbb{R}^d$ corrupted with Gaussian noise of variance $\delta$ to within $\varepsilon^2$ in KL divergence. Our proof extends the Girsanov-based methods of previous works. We introduce a refined treatment of the error from discretizing the reverse SDE inspired by stochastic localization.

In light of the widespread success of generative models, a significant amount of research has gone into speeding up their sampling time. However, generative models are often sampled multiple times to obtain a diverse set incurring a cost that is orthogonal to sampling time. We tackle the question of how to improve diversity and sample efficiency by moving beyond the common assumption of independent samples. We propose particle guidance, an extension of diffusion-based generative sampling where a joint-particle time-evolving potential enforces diversity. We analyze theoretically the joint distribution that particle guidance generates, how to learn a potential that achieves optimal diversity, and the connections with methods in other disciplines. Empirically, we test the framework both in the setting of conditional image generation, where we are able to increase diversity without affecting quality, and molecular conformer generation, where we reduce the state-of-the-art median error by 13% on average.

Posterior sampling has been shown to be a powerful Bayesian approach for solving imaging inverse problems. The recent plug-and-play unadjusted Langevin algorithm (PnP-ULA) has emerged as a promising method for Monte Carlo sampling and minimum mean squared error (MMSE) estimation by combining physical measurement models with deep-learning priors specified using image denoisers. However, the intricate relationship between the sampling distribution of PnP-ULA and the mismatched data-fidelity and denoiser has not been theoretically analyzed. We address this gap by proposing a posterior-$L_2$ pseudometric and using it to quantify an explicit error bound for PnP-ULA under mismatched posterior distribution. We numerically validate our theory on several inverse problems such as sampling from Gaussian mixture models and image deblurring. Our results suggest that the sensitivity of the sampling distribution of PnP-ULA to a mismatch in the measurement model and the denoiser can be precisely characterized.

Denoising diffusion models have become ubiquitous for generative modeling. The core idea is to transport the data distribution to a Gaussian by using a diffusion. Approximate samples from the data distribution are then obtained by estimating the time-reversal of this diffusion using score matching ideas. We follow here a similar strategy to sample from unnormalized probability densities and compute their normalizing constants. However, the time-reversed diffusion is here simulated by using an original iterative particle scheme relying on a novel score matching loss. Contrary to standard denoising diffusion models, the resulting Particle Denoising Diffusion Sampler (PDDS) provides asymptotically consistent estimates under mild assumptions. We demonstrate PDDS on multimodal and high dimensional sampling tasks.

Mass transport problems arise in many areas of machine learning whereby one wants to compute a map transporting one distribution to another. Generative modeling techniques like Generative Adversarial Networks (GANs) and Denoising Diffusion Models (DMMs) have been successfully adapted to solve such transport problems, resulting in CycleGAN and Bridge Matching respectively. However, these methods do not approximate Optimal Transport (OT) maps, which are known to have desirable properties. Existing techniques approximating OT maps for high-dimensional data-rich problems, including DDMs-based Rectified Flow and Schrodinger bridge procedures, require fully training a DDM-type model at each iteration, or use mini-batch techniques which can introduce significant errors. We propose a novel algorithm to compute the Schrodinger bridge, a dynamic entropy-regularized version of OT, that eliminates the need to train multiple DDMs-like models. This algorithm corresponds to a discretization of a flow of path measures, referred to as the Schrodinger Bridge Flow, whose only stationary point is the Schrodinger bridge. We demonstrate the performance of our algorithm on a variety of unpaired data translation tasks.

The design of novel protein structures remains a challenge in protein engineering for applications across biomedicine and chemistry. In this line of work, a diffusion model over rigid bodies in 3D (referred to as frames) has shown success in generating novel, functional protein backbones that have not been observed in nature. However, there exists no principled methodological framework for diffusion on SE(3), the space of orientation preserving rigid motions in R3, that operates on frames and confers the group invariance. We address these shortcomings by developing theoretical foundations of SE(3) invariant diffusion models on multiple frames followed by a novel framework, FrameDiff, for estimating the SE(3) equivariant score over multiple frames. We apply FrameDiff on monomer backbone generation and find it can generate designable monomers up to 500 amino acids without relying on a pretrained protein structure prediction network that has been integral to previous methods. We find our samples are capable of generalizing beyond any known protein structure.

Solving transport problems, i.e. finding a map transporting one given distribution to another, has numerous applications in machine learning. Novel mass transport methods motivated by generative modeling have recently been proposed, e.g. Denoising Diffusion Models (DDMs) and Flow Matching Models (FMMs) implement such a transport through a Stochastic Differential Equation (SDE) or an Ordinary Differential Equation (ODE). However, while it is desirable in many applications to approximate the deterministic dynamic Optimal Transport (OT) map which admits attractive properties, DDMs and FMMs are not guaranteed to provide transports close to the OT map. In contrast, Schrödinger bridges (SBs) compute stochastic dynamic mappings which recover entropy-regularized versions of OT. Unfortunately, existing numerical methods approximating SBs either scale poorly with dimension or accumulate errors across iterations. In this work, we introduce Iterative Markovian Fitting (IMF), a new methodology for solving SB problems, and Diffusion Schrödinger Bridge Matching (DSBM), a novel numerical algorithm for computing IMF iterates. DSBM significantly improves over previous SB numerics and recovers as special/limiting cases various recent transport methods. We demonstrate the performance of DSBM on a variety of problems.

We propose a new class of generative model that naturally handles data of varying dimensionality by jointly modeling the state and dimension of each datapoint. The generative process is formulated as a jump diffusion process that makes jumps between different dimensional spaces. We first define a dimension destroying forward noising process, before deriving the dimension creating time-reversed generative process along with a novel evidence lower bound training objective for learning to approximate it. Simulating our learned approximation to the time-reversed generative process then provides an effective way of sampling data of varying dimensionality by jointly generating state values and dimensions. We demonstrate our approach on molecular and video datasets of varying dimensionality, reporting better compatibility with test-time diffusion guidance imputation tasks and improved interpolation capabilities versus fixed dimensional models that generate state values and dimensions separately.

Denoising diffusion models have recently emerged as the predominant paradigm for generative modelling on image domains. In addition, their extension to Riemannian manifolds has facilitated a range of applications across the natural sciences. While many of these problems stand to benefit from the ability to specify arbitrary, domain-informed constraints, this setting is not covered by the existing (Riemannian) diffusion model methodology. Recent work has attempted to address this issue by constructing novel noising processes based on the reflected Brownian motion and logarithmic barrier methods. However, the associated samplers are either computationally burdensome or only apply to convex subsets of Euclidean space. In this paper, we introduce an alternative, simple noising scheme based on Metropolis sampling that affords substantial gains in computational efficiency and empirical performance compared to the earlier samplers. Of independent interest, we prove that this new process corresponds to a valid discretisation of the reflected Brownian motion. We demonstrate the scalability and flexibility of our approach on a range of problem settings with convex and non-convex constraints, including applications from geospatial modelling, robotics and protein design.

Denoising diffusion models have proven to be a flexible and effective paradigm for generative modelling. Their recent extension to infinite dimensional Euclidean spaces has allowed for the modelling of stochastic processes. However, many problems in the natural sciences incorporate symmetries and involve data living in non-Euclidean spaces. In this work, we extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We do so by a) constructing a noising process which admits, as limiting distribution, a geometric Gaussian process that transforms under the symmetry group of interest, and b) approximating the score with a neural network that is equivariant w.r.t. this group. We show that with these conditions, the generative functional model admits the same symmetry. We demonstrate scalability and capacity of the model, using a novel Langevin-based conditional sampler, to fit complex scalar and vector fields, with Euclidean and spherical codomain, on synthetic and real-world weather data.

In this paper, we propose Barrier Hamiltonian Monte Carlo (BHMC), a version of the HMC algorithm which aims at sampling from a Gibbs distribution $\pi$ on a manifold $\mathsf{M}$, endowed with a Hessian metric $\mathfrak{g}$ derived from a self-concordant barrier. Our method relies on Hamiltonian dynamics which comprises $\mathfrak{g}$. Therefore, it incorporates the constraints defining $\mathsf{M}$ and is able to exploit its underlying geometry. However, the corresponding Hamiltonian dynamics is defined via non separable Ordinary Differential Equations (ODEs) in contrast to the Euclidean case. It implies unavoidable bias in existing generalization of HMC to Riemannian manifolds. In this paper, we propose a new filter step, called ``involution checking step'', to address this problem. This step is implemented in two versions of BHMC, coined continuous BHMC (c-bHMC) and numerical BHMC (n-BHMC) respectively. Our main results establish that these two new algorithms generate reversible Markov chains with respect to $\pi$ and do not suffer from any bias in comparison to previous implementations. Our conclusions are supported by numerical experiments where we consider target distributions defined on polytopes.

Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at minimizing the integral of a cost function with respect to a distribution with some prescribed marginals. In this paper, we consider an entropic version of mOT with a tree-structured quadratic cost, i.e., a function that can be written as a sum of pairwise cost functions between the nodes of a tree. To address this problem, we develop Tree-based Diffusion Schr\"odinger Bridge (TreeDSB), an extension of the Diffusion Schr\"odinger Bridge (DSB) algorithm. TreeDSB corresponds to a dynamic and continuous state-space counterpart of the multimarginal Sinkhorn algorithm. A notable use case of our methodology is to compute Wasserstein barycenters which can be recast as the solution of a mOT problem on a star-shaped tree. We demonstrate that our methodology can be applied in high-dimensional settings such as image interpolation and Bayesian fusion.

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance.Score-based generative modelling (SGM) consists of a noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails adenoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here \emph{Riemannian Score-based Generative Models} (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of compact manifolds, and in particular with earth and climate science spherical data.

We provide the first complete continuous time framework for denoising diffusion models of discrete data. This is achieved by formulating the forward noising process and corresponding reverse time generative process as Continuous Time Markov Chains (CTMCs). The model can be efficiently trained using a continuous time version of the ELBO. We simulate the high dimensional CTMC using techniques developed in chemical physics and exploit our continuous time framework to derive high performance samplers that we show can outperform discrete time methods for discrete data. The continuous time treatment also enables us to derive a novel theoretical result bounding the error between the generated sample distribution and the true data distribution.

Score-based generative models (SGMs) synthesize new data samples from Gaussian white noise by running a time-reversed Stochastic Differential Equation (SDE) whose drift coefficient depends on some probabilistic score. The discretization of such SDEs typically requires a large number of time steps and hence a high computational cost. This is because of ill-conditioning properties of the score that we analyze mathematically. Previous approaches have relied on multiscale generation to considerably accelerate SGMs. We explain how this acceleration results from an implicit factorization of the data distribution into a product of conditional probabilities of wavelet coefficients across scales. The resulting Wavelet Score-based Generative Model (WSGM) synthesizes wavelet coefficients with the same number of time steps at all scales, and its time complexity therefore grows linearly with the image size. This is proved mathematically for Gaussian distributions, and shown numerically for physical processes at phase transition and natural image datasets.

Many generative models synthesize data by transforming a standard Gaussian random variable using a deterministic neural network. Among these models are the Variational Autoencoders and the Generative Adversarial Networks. In this work, we call them "push-forward" models and study their expressivity. We formally demonstrate that the Lipschitz constant of these generative networks has to be large in order to fit multimodal distributions. More precisely, we show that the total variation distance and the Kullback-Leibler divergence between the generated and the data distribution are bounded from below by a constant depending on the mode separation and the Lipschitz constant. Since constraining the Lipschitz constants of neural networks is a common way to stabilize generative models, there is a provable trade-off between the ability of push-forward models to approximate multimodal distributions and the stability of their training. We validate our findings on one-dimensional and image datasets and empirically show that the recently introduced diffusion models do not suffer of such limitation.

Denoising diffusion models have recently emerged as a powerful class of generative models. They provide state-of-the-art results, not only for unconditional simulation, but also when used to solve conditional simulation problems arising in a wide range of inverse problems. A limitation of these models is that they are computationally intensive at generation time as they require simulating a diffusion process over a long time horizon. When performing unconditional simulation, a Schr{ö}dinger bridge formulation of generative modeling leads to a theoretically grounded algorithm shortening generation time which is complementary to other proposed acceleration techniques. We extend the Schrödinger bridge framework to conditional simulation. We demonstrate this novel methodology on various applications including image super-resolution, optimal filtering for state-space models and the refinement of pre-trained networks. Our code can be found at https://github.com/vdeborto/cdsb.

Since the seminal work of Venkatakrishnan et al. in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. The PnP algorithms proposed in the literature mainly differ in the iterative schemes they use for optimisation or for sampling. In the case of optimisation schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a MAP estimate. In the case of sampling schemes, to the best of our knowledge, there is no known proof of convergence. There also remain important open questions regarding whether the underlying Bayesian models and estimators are well defined, well-posed, and have the basic regularity properties required to support these numerical schemes. To address these limitations, this paper develops theory, methods, and provably convergent algorithms for performing Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA (Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Stochastic Gradient Descent) for MAP inference. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for these two algorithms under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well-posed. The proposed algorithms are demonstrated on several canonical problems such as image deblurring, inpainting, and denoising, where they are used for point estimation as well as for uncertainty visualisation and quantification.

This paper proposes a thorough theoretical analysis of Stochastic Gradient Descent (SGD) with non-increasing step sizes. First, we show that the recursion defining SGD can be provably approximated by solutions of a time inhomogeneous Stochastic Differential Equation (SDE) using an appropriate coupling. In the specific case of a batch noise we refine our results using recent advances in Stein’s method. Then, motivated by recent analyses of deterministic and stochastic optimization methods by their continuous counterpart, we study the long-time behavior of the continuous processes at hand and establish non-asymptotic bounds. To that purpose, we develop new comparison techniques which are of independent interest. Adapting these techniques to the discrete setting, we show that the same results hold for the corresponding SGD sequences. In our analysis, we notably improve non-asymptotic bounds in the convex setting for SGD under weaker assumptions than the ones considered in previous works. Finally, we also establish finite-time convergence results under various conditions, including relaxations of the famous Ł{ojasciewicz} inequality, which can be applied to a class of non-convex functions.

Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is given by a Stochastic Differential Equation (SDE), Song et al (2021) demonstrate how the time inhomogeneous drift of the associated reverse-time SDE may be estimated using score-matching. A limitation of this approach is that the forward-time SDE must be run for a sufficiently long time for the final distribution to be approximately Gaussian. In contrast, solving the Schrödinger Bridge (SB) problem, i.e. an entropy-regularized optimal transport problem on path spaces, yields diffusions which generate samples from the data distribution in finite time. We present Diffusion SB (DSB), an original approximation of the Iterative Proportional Fitting (IPF) procedure to solve the SB problem, and provide theoretical analysis along with generative modeling experiments. The first DSB iteration recovers the methodology proposed by Song et al. (2021), with the flexibility of using shorter time intervals, as subsequent DSB iterations reduce the discrepancy between the final-time marginal of the forward (resp. backward) SDE with respect to the prior (resp. data) distribution. Beyond generative modeling, DSB offers a widely applicable computational optimal transport tool as the continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi, 2013).

Approximate Bayesian Computation (ABC) is a popular method for approximate inference in generative models with intractable but easy-to-sample likelihood. It constructs an approximate posterior distribution by finding parameters for which the simulated data are close to the observations in terms of summary statistics. These statistics are defined beforehand and might induce a loss of information, which has been shown to deteriorate the quality of the approximation. To overcome this problem, Wasserstein-ABC has been recently proposed, and compares the datasets via the Wasserstein distance between their empirical distributions, but does not scale well to the dimension or the number of samples. We propose a new ABC technique, called Sliced-Wasserstein ABC and based on the Sliced-Wasserstein distance, which has better computational and statistical properties. We derive two theoretical results showing the asymptotical consistency of our approach, and we illustrate its advantages on synthetic data and an image denoising task.

In this paper, we investigate the limiting behavior of a continuous-time counterpart of the Stochastic Gradient Descent (SGD) algorithm applied to two-layer overparameterized neural networks, as the number or neurons (i.e., the size of the hidden layer) $N \to \plusinfty$. Following a probabilistic approach, we show `propagation of chaos' for the particle system defined by this continuous-time dynamics under different scenarios, indicating that the statistical interaction between the particles asymptotically vanishes. In particular, we establish quantitative convergence with respect to $N$ of any particle to a solution of a mean-field McKean-Vlasov equation in the metric space endowed with the Wasserstein distance. In comparison to previous works on the subject, we consider settings in which the sequence of stepsizes in SGD can potentially depend on the number of neurons and the iterations. We then identify two regimes under which different mean-field limits are obtained, one of them corresponding to an implicitly regularized version of the minimization problem at hand. We perform various experiments on real datasets to validate our theoretical results, assessing the existence of these two regimes on classification problems and illustrating our convergence results.

This paper presents a detailed theoretical analysis of the three stochastic approximation proximal gradient algorithms proposed in our companion paper [A. F. Vidal et al., SIAM J. Imaging Sci., 13 (2020), pp. 1945--1989] to set regularization parameters by marginal maximum likelihood estimation. We prove the convergence of a more general stochastic approximation scheme that includes the three algorithms of [A. F. Vidal et al., SIAM J. Imaging Sci., 13 (2020), pp. 1945--1989] as special cases. This includes asymptotic and nonasymptotic convergence results with natural and easily verifiable conditions, as well as explicit bounds on the convergence rates. Importantly, the theory is also general in that it can be applied to other intractable optimization problems. A main novelty of the work is that the stochastic gradient estimates of our scheme are constructed from inexact proximal Markov chain Monte Carlo samplers. This allows the use of samplers that scale efficiently to large problems and for which we have precise theoretical guarantees.

Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularizing the estimation problem to make it well- posed. This often requires setting the value of the so-called regularization parameters that control the amount of regularization enforced. These parameters are notoriously difficult to set a priori and can have a dramatic impact on the recovered estimates. In this work, we propose a general empirical Bayesian method for setting regularization parameters in imaging problems that are convex w.r.t. the unknown image. Our method calibrates regularization parameters directly from the observed data by maximum marginal likelihood estimation and can simultaneously estimate multiple regularization parameters. Furthermore, the proposed algorithm uses the same basic operators as proximal optimization algorithms, namely gradient and proximal operators, and it is therefore straightforward to apply to problems that are currently solved by using proximal optimization techniques. Our methodology is demonstrated with a range of experiments and comparisons with alternative approaches from the literature. The considered experiments include image denoising, nonblind image deconvolution, and hyperspectral unmixing, using synthesis and analysis priors involving the $\ell_1$, total-variation, total-variation and $\ell_1$, and total-generalized-variation pseudonorms. A detailed theoretical analysis of the proposed method is presented in our companion paper [V. De Bortoli et al., SIAM J. Imaging Sci., 13 (2020), pp. 1990--2028].

In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally. To do so, we define an auto-similarity function which, given one image, computes a dissimilarity measurement between patches. To derive a criterion for taking a decision on the similarity between two patches, we present an a contrario model. Namely, two patches are said to be similar if the associated dissimilarity measurement is unlikely to happen in a background model. Choosing Gaussian random fields as background models, we derive nonasymptotic expressions for the probability distribution function of similarity measurements. We present an algorithm in order to assess redundancy in natural images and discuss applications in denoising, periodicity analysis, and texture ranking.

Wavelet‐based segmentation approaches are widely used for texture segmentation purposes because of their ability to characterise different textures. In this study, the authors assess the influence of the chosen wavelet and propose to use the recently introduced empirical wavelets. We show that the adaptability of the empirical wavelet permits to reach better results than classic wavelets. To focus only on the textural information, they also propose to perform a cartoon + texture decomposition step before applying the segmentation algorithm. The proposed method is tested on six classic benchmarks, based on several popular texture images.