arXiv subjects

Agnès Desolneux

Publications and source records attributed to Agnès Desolneux.

15 recordsLinked to original sources

arXiv2026

Differentiable Expectation-Maximisation and Applications to Gaussian Mixture Model Optimal Transport

The Expectation-Maximisation (EM) algorithm is a central tool in statistics and machine learning, widely used for latent-variable models such as Gaussian Mixture Models (GMMs). Despite its ubiquity, EM is typically treated as a non-differentiable black box, preventing its integration into modern learning pipelines where end-to-end gradient propagation is essential. In this work, we present and compare several differentiation strategies for EM, from full automatic differentiation to approximate methods, assessing their accuracy and computational efficiency. As a key application, we leverage this differentiable EM in the computation of the Mixture Wasserstein distance $\mathrm{MW}_2$ between GMMs, allowing $\mathrm{MW}_2$ to be used as a differentiable loss in imaging and machine learning tasks. To complement our practical use of $\mathrm{MW}_2$, we contribute a novel stability result which provides theoretical justification for the use of $\mathrm{MW}_2$ with EM, and also introduce a novel unbalanced variant of $\mathrm{MW}_2$. Numerical experiments on barycentre computation, colour and style transfer, image generation, and texture synthesis illustrate the versatility of the proposed approach in different settings.

cs.LG↗

arXiv2026

On quantitative Laplace-type convergence results for some exponential probability measures, with two applications

Laplace-type results characterize the limit of sequence of measures $(π_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} π_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$. If a limiting distribution $π_0$ exists, it concentrates on the minimizers of the potential $U$. Classical results require the invertibility of the Hessian of $U$ in order to establish such asymptotics. In this work, we study the particular case of norm-like potentials $U$ and establish quantitative bounds between $π_\varepsilon$ and $π_0$ w.r.t. the Wasserstein distance of order $1$ under an invertibility condition of a generalized Jacobian. One key element of our proof is the use of geometric measure theory tools such as the coarea formula. We apply our results to the study of maximum entropy models (microcanonical/macrocanonical distributions) and to the convergence of the iterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at low temperatures for non-convex minimization.

math.PR↗

arXiv2026

Repulsive Monte Carlo on the sphere for the sliced Wasserstein distance

In this paper, we consider the problem of computing the integral of a function on the unit sphere, in any dimension, using Monte Carlo methods. Although the methods we present are general, our guiding thread is the sliced Wasserstein distance between two measures on $\mathbb{R}^d$, which is precisely an integral on the $d$-dimensional sphere. The sliced Wasserstein distance (SW) has gained momentum in machine learning either as a proxy to the less computationally tractable Wasserstein distance, or as a distance in its own right, due in particular to its built-in alleviation of the curse of dimensionality. There has been recent numerical benchmarks of quadratures for the sliced Wasserstein, and our viewpoint differs in that we concentrate on quadratures where the nodes are repulsive, i.e. negatively dependent. Indeed, negative dependence can bring variance reduction when the quadrature is adapted to the integration task. Our first contribution is to extract and motivate quadratures from the recent literature on determinantal point processes (DPPs) and repelled point processes, as well as repulsive quadratures from the literature specific to the sliced Wasserstein distance. We then numerically benchmark these quadratures. Moreover, we analyze the variance of the UnifOrtho estimator, an orthogonal Monte Carlo estimator. Our analysis sheds light on UnifOrtho's success for the estimation of the sliced Wasserstein in large dimensions, as well as counterexamples from the literature. Our final recommendation for the computation of the sliced Wasserstein distance is to use randomized quasi-Monte Carlo in low dimensions and UnifOrtho in large dimensions. DPP-based quadratures only shine when quasi-Monte Carlo also does, while repelled quadratures show moderate variance reduction in general, but more theoretical effort is needed to make them robust.

stat.ML↗

arXiv2025

Constrained Approximate Optimal Transport Maps

We investigate finding a map $g$ within a function class $G$ that minimises an Optimal Transport (OT) cost between a target measure $ν$ and the image by $g$ of a source measure $μ$. This is relevant when an OT map from $μ$ to $ν$ does not exist or does not satisfy the desired constraints of $G$. We address existence and uniqueness for generic subclasses of $L$-Lipschitz functions, including gradients of (strongly) convex functions and typical Neural Networks. We explore a variant that approaches a transport plan, showing equivalence to a map problem in some cases. For the squared Euclidean cost, we propose alternating minimisation over a transport plan $π$ and map $g$, with the optimisation over $g$ being the $L^2$ projection on $G$ of the barycentric mapping $\overlineπ$. In dimension one, this global problem equates the $L^2$ projection of $\overline{π^*}$ onto $G$ for an OT plan $π^*$ between $μ$ and $ν$, but this does not extend to higher dimensions. We introduce a simple kernel method to find $g$ within a Reproducing Kernel Hilbert Space in the discrete case. We present numerical methods for $L$-Lipschitz gradients of $\ell$-strongly convex potentials, and study the convergence of Stochastic Gradient Descent methods for Neural Networks. We finish with an illustration on colour transfer, applying learned maps on new images, and showcasing outlier robustness.

math.OC↗

arXiv2025

Bridging Contrastive Learning and Domain Adaptation: Theoretical Perspective and Practical Application

This work studies the relationship between Contrastive Learning and Domain Adaptation from a theoretical perspective. The two standard contrastive losses, NT-Xent loss (Self-supervised) and Supervised Contrastive loss, are related to the Class-wise Mean Maximum Discrepancy (CMMD), a dissimilarity measure widely used for Domain Adaptation. Our work shows that minimizing the contrastive losses decreases the CMMD and simultaneously improves class-separability, laying the theoretical groundwork for the use of Contrastive Learning in the context of Domain Adaptation. Due to the relevance of Domain Adaptation in medical imaging, we focused the experiments on mammography images. Extensive experiments on three mammography datasets - synthetic patches, clinical (real) patches, and clinical (real) images - show improved Domain Adaptation, class-separability, and classification performance, when minimizing the Supervised Contrastive loss.

cs.LG↗

arXiv2024

BN-SCAFFOLD: controlling the drift of Batch Normalization statistics in Federated Learning

Federated Learning (FL) is gaining traction as a learning paradigm for training Machine Learning (ML) models in a decentralized way. Batch Normalization (BN) is ubiquitous in Deep Neural Networks (DNN), as it improves convergence and generalization. However, BN has been reported to hinder performance of DNNs in heterogeneous FL. Recently, the FedTAN algorithm has been proposed to mitigate the effect of heterogeneity on BN, by aggregating BN statistics and gradients from all the clients. However, it has a high communication cost, that increases linearly with the depth of the DNN. SCAFFOLD is a variance reduction algorithm, that estimates and corrects the client drift in a communication-efficient manner. Despite its promising results in heterogeneous FL settings, it has been reported to underperform for models with BN. In this work, we seek to revive SCAFFOLD, and more generally variance reduction, as an efficient way of training DNN with BN in heterogeneous FL. We introduce a unified theoretical framework for analyzing the convergence of variance reduction algorithms in the BN-DNN setting, inspired of by the work of Wang et al. 2023, and show that SCAFFOLD is unable to remove the bias introduced by BN. We thus propose the BN-SCAFFOLD algorithm, which extends the client drift correction of SCAFFOLD to BN statistics. We prove convergence using the aforementioned framework and validate the theoretical results with experiments on MNIST and CIFAR-10. BN-SCAFFOLD equals the performance of FedTAN, without its high communication cost, outperforming Federated Averaging (FedAvg), SCAFFOLD, and other FL algorithms designed to mitigate BN heterogeneity.

cs.LG↗

arXiv2024

Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space

The Gromov-Wasserstein (GW) distance is frequently used in machine learning to compare distributions across distinct metric spaces. Despite its utility, it remains computationally intensive, especially for large-scale problems. Recently, a novel Wasserstein distance specifically tailored for Gaussian mixture models (GMMs) and known as MW2 (mixture Wasserstein) has been introduced by several authors. In scenarios where data exhibit clustering, this approach simplifies to a small-scale discrete optimal transport problem, which complexity depends solely on the number of Gaussian components in the GMMs. This paper aims to incorporate invariance properties into MW2. This is done by introducing new Gromov-type distances, designed to be isometry-invariant in Euclidean spaces and applicable for comparing GMMs across different dimensional spaces. Our first contribution is the Mixture Gromov Wasserstein distance (MGW2), which can be viewed as a "Gromovized" version of MW2. This new distance has a straightforward discrete formulation, making it highly efficient for estimating distances between GMMs in practical applications. To facilitate the derivation of a transport plan between GMMs, we present a second distance, the Embedded Wasserstein distance (EW2). This distance turns out to be closely related to several recent alternatives to Gromov-Wasserstein. We show that EW2 can be adapted to derive a distance as well as optimal transportation plans between GMMs. We demonstrate the efficiency of these newly proposed distances on medium to large-scale problems, including shape matching and hyperspectral image color transfer.

stat.ML↗

arXiv2022

Can Push-forward Generative Models Fit Multimodal Distributions?

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 show 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 generative models consisting of stacked networks with stochastic input at each step, such as diffusion models do not suffer of such limitations.

stat.ML↗

arXiv2021

Gromov-Wasserstein Distances between Gaussian Distributions

The Gromov-Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. In this paper, we focus on the Gromov-Wasserstein distance with a ground cost defined as the squared Euclidean distance and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also solution of the linear Gromov-Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov-Wasserstein distance between Gaussian distributions.

math.PR↗

arXiv2021

Exact Sampling of Determinantal Point Processes without Eigendecomposition

Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel $K$ that can be seen as a matrix storing the similarity between points. The diversity comes from the fact that the inclusion probability of a subset is equal to the determinant of a submatrice of $K$. The exact algorithm to sample DPPs uses the spectral decomposition of $K$, a computation that becomes costly when dealing with a high number of points. Here, we present an alternative exact algorithm in the discrete setting that avoids the eigenvalues and the eigenvectors computation. Instead, it relies on Cholesky decompositions. This is a two steps strategy: first, it samples a Bernoulli point process with an appropriate distribution, then it samples the target DPP distribution through a thinning procedure. Not only is the method used here innovative, but this algorithm can be competitive with the original algorithm or even faster for some applications specified here.

stat.ML↗

arXiv2020

Unidentified Floating Object detection in maritime environment using dictionary learning

Maritime domain is one of the most challenging scenarios for object detection due to the complexity of the observed scene. In this article, we present a new approach to detect unidentified floating objects in the maritime environment. The proposed approach is capable of detecting floating objects without any prior knowledge of their visual appearance, shape or location. The input image from the video stream is denoised using a visual dictionary learned from a K-SVD algorithm. The denoised image is made of self-similar content. Later, we extract the residual image, which is the difference between the original image and the denoised (self-similar) image. Thus, the residual image contains noise and salient structures (objects). These salient structures can be extracted using an a contrario model. We demonstrate the capabilities of our algorithm by testing it on videos exhibiting varying maritime scenarios.

cs.CV↗

arXiv2017

A survey of exemplar-based texture synthesis

Exemplar-based texture synthesis is the process of generating, from an input sample, new texture images of arbitrary size and which are perceptually equivalent to the sample. The two main approaches are statistics-based methods and patch re-arrangement methods. In the first class, a texture is characterized by a statistical signature; then, a random sampling conditioned to this signature produces genuinely different texture images. The second class boils down to a clever "copy-paste" procedure, which stitches together large regions of the sample. Hybrid methods try to combine ideas from both approaches to avoid their hurdles. The recent approaches using convolutional neural networks fit to this classification, some being statistical and others performing patch re-arrangement in the feature space. They produce impressive synthesis on various kinds of textures. Nevertheless, we found that most real textures are organized at multiple scales, with global structures revealed at coarse scales and highly varying details at finer ones. Thus, when confronted with large natural images of textures the results of state-of-the-art methods degrade rapidly, and the problem of modeling them remains wide open.

cs.CV↗

arXiv2012

Crossings of smooth shot noise processes

In this paper, we consider smooth shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the mean number of crossings function is obtained through an integral formula. Moreover, as the intensity increases, or equivalently, as the number of shots becomes larger, a normal convergence to the classical Rice's formula for Gaussian processes is obtained. The Gaussian kernel function, that corresponds to many applications in physics, is studied in detail and two different regimes are exhibited.

math.PR↗

arXiv2006

A zero-one law for first-order logic on random images

For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or 1, provided both $p(n)n^{\frac{2}{k}}$ and $(1-p(n))n^{\frac{2}{k}}$ tend to 0 or $+\infty$, for any integer $k$. The result is proved by computing the threshold function for basic local sentences, and applying Gaifman's theorem.

math.PR↗

arXiv2006

Image denoising by statistical area thresholding

Area openings and closings are morphological filters which efficiently suppress impulse noise from an image, by removing small connected components of level sets. The problem of an objective choice of threshold for the area remains open. Here, a mathematical model for random images will be considered. Under this model, a Poisson approximation for the probability of appearance of any local pattern can be computed. In particular, the probability of observing a component with size larger than $k$ in pure impulse noise has an explicit form. This permits the definition of a statistical test on the significance of connected components, thus providing an explicit formula for the area threshold of the denoising filter, as a function of the impulse noise probability parameter. Finally, using threshold decomposition, a denoising algorithm for grey level images is proposed.

math.PR↗