Séminaire Images Optimisation et Probabilités

Responsable : Luis Fredes et Camille Male

Le 11 janvier 2024 à 11:00

Salle de conférénces

Valentin Debarnot University of Basel

Algorithmes de deep-learning pour la reconstruction en microscopie électronique et à fluorescence.

Dans cette présentation, je présenterai différents aspects qui limitent la résolution en microscopie électronique et à fluorescence. Après avoir défini ces problèmes dans un formalisme de problème inverse, j'introduirai différents outils qui nous ont permis d'atténuer certaines limitations, et je discuterais différentes pistes de recherche possibles pour prendre en compte les limitations restantes. J'utiliserais des outils de machine learning (e.g. réseau de neurones implicites, deep image prior, dérivation automatique) pour résoudre des problèmes inverses en microscopie.

Le 25 janvier 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférénces

Paul Catala University of Osnabrückv

Trigonometric approximations of the sparse super-resolution problem in Wasserstein distances

In this talk, I will discuss the recovery of an arbitrary measure on the $d$-dimensional torus, given trigonometric moments up to degree $n$. Considering the convolution of the measure with powers of the Fejér kernel, which can be computed efficiently from the truncated moment sequence, I will provide rates of convergence of the resulting polynomial density towards the measure in the $p$-Wasserstein distance, as the degree $n$ increases. In particular, I will show that the best possible rate for polynomial approximation is inversely proportional to the degree, and that it is achieved by adequately choosing the power to which the kernel is raised. Finally, I will discuss another class of polynomial approximations, similar although not based on convolution, that converge pointwise to the characteristic function of the support of the measure. This is joint work with Mathias Hockmann, Stefan Kunis and Markus Wageringel.

Le 1er février 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférénces

Jamal Najim Université Gustave Eiffel

Equilibres de grands systèmes de Lotka-Volterra couplés par des matrices aléatoires non-hermitiennes

Les systèmes de Lotka-Volterra sont des équations différentielles couplées par une matrice dite d’interactions. On s’intéressera au cas où la matrice d’interactions est une grande matrice aléatoire, modèle fréquemment utilisé en écologie théorique pour comprendre les réseaux trophiques. Dans les cas d’existence d’un équilibre stable, aléatoire par nature, on s’attachera à décrire certaines propriétés statistiques de cet équilibre, comme par exemple la proportion des composantes non nulles. On s’intéressera à des modèles matriciels non-hermitiens, de type Ginibre réel et plus généralement elliptique, et on montrera comment des algorithme de type AMP (Approximate Message Passing) permettent d’accéder aux propriétés statistiques de ces équilibres.

Travail en collaboration avec Y. Gueddari et W. Hachem, voir aussi https://arxiv.org/abs/2302.07820

Le 8 février 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de Conférences

Bastien Laville INRIA

Some developments on off-the-grid curve reconstruction: divergence regularisation and untangling by Riemannian metric

Recent years have seen the development of super-resolution variational optimisation in measure spaces. These so-called off-the-grid approaches offer both theoretical and numerical results, with very convincing results in biomedical imaging. However, the gridless variational optimisation is generally formulated for reconstruction of point sources, which is not always suitable for biomedical imaging applications: more realistic biological structures such as curves should also be reconstructed. In the first part of this talk, we propose a new strategy for the reconstruction of curves in an image through an off-the-grid variational framework, thanks to the sharp characterisation of the extreme points of the unit ball of a new regulariser thus enabling new theoretical and numerical results for optical imaging. In a second part of the talk, we investigate a new strategy for off-the-grid curve untangling, with some practical results for Localisation Microscopy.

Le 15 février 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférénces

Sebastien Herbreteau EPFL

Towards better conditioned and interpretable neural networks : a study of the normalization-equivariance property

In many information processing systems, it may be desirable to ensure that any change in the input, whether by shifting or scaling, results in a corresponding change in the system response. While deep neural networks are gradually replacing all traditional automatic processing methods, they surprisingly do not guarantee such normalization-equivariance (scale & shift) property, which can be detrimental in many applications. Inspired by traditional methods in image denoising, we propose a methodology to adapt existing convolutional neural networks so that normalization-equivariance holds by design and without performance loss. Our main claim is that not only ordinary unconstrained convolutional layers, but also all activation functions, including the ReLU (rectified linear unit), which are applied element-wise to the pre-activated neurons, should be completely removed from neural networks and replaced by better conditioned alternatives. As a result, we show that better conditioning improves the interpretability but also the robustness of these networks to outliers, which is experimentally confirmed in the context of image denoising.

Le 7 mars 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférénces

Radu-Alexandru Dragomir Telecom Paris

Quartic Optimization Problems

Many tasks in signal processing and machine learning involve minimizing polynomials of degree four. These include phase retrieval, matrix factorization, sensor network localization and many more. In this talk, I will give an overview of the challenges of quartic minimization as well as some complexity results. In particular, we will focus on a particular class of convex quartic problems, and we analyze the notion of quartic condition number. We design an algorithm for reducing this condition number. To do so, we build a preconditioner using a generalized version of the Lewis weights (a.k.a leverage scores), and we show that it is optimal in some specific sense.

Based on joint work with Yurii Nesterov.

Le 14 mars 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférences

Jianyu Ma Université de Toulouse

Displacement functional and absolute continuity of Wasserstein barycenters.

Barycenters are defined to average probability measures on metric spaces.
For Wasserstein spaces, (Wasserstein) barycenter is a direct generalization of the celebrated McCann interpolation, which corresponds to the barycenters of measures $\lambda \delta_{\mu_1} + (1 - \lambda) \delta_{\mu_2}$.
In the talk, we consider Wasserstein barycenters on Riemannian manifolds,
and discuss the displacement functional used by the author in arXiv:2310.13832 to prove their absolute continuity with lower Ricci curvature bound assumptions.
It is different from the widely used displacement convexity property combined with gradient flow, but still manifests an intriguing connection with the curvature-dimension condition.
If time allowed, we will also explain how the Souslin space theory is applied in the proof,
which is an unexpected technique for optimal transport.

Le 21 mars 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférénces.

Laura Girometti et Léo Portales Université de Bologne et de Toulouse

Deux exposés

Title Léo : Convergence of the iterates of Lloyd's algorithm
Summary : The finding of a discrete measure that approaches a target density, called quantization, is an important aspect of machine learning and is usually done using Lloyd’s algorithm; a continuous counterpart to the K-means algorithm. We have studied two variants of this algorithm: one where we specify the former measure to be uniform (uniform quantization) and one where the weights associated to each point is adjusted to fit the target density (optimal quantization). In either case it is not yet known in the literature whether the iterates of these algorithms converge simply. We proved so with the assumption that the target density is analytic and restricted to a semi algebraic compact and convex set. We do so using tools from o-minimal geometry as well as the Kurdyka-Lojasiewicz inequality. We also proved along the way the definability in an o-minimal structure of functions of the form Y := (y1, ..., yN ) → D(\mu,1/N sum_{i=1}^N \delta_{y_i}) for the following divergences D: the general Wp Wasserstein distance, the max-sliced Wasserstein distance and the entropic regularized Wasserstein distance.

Title Laura : Quaternary Image Decomposition
Summary : Decomposing an image into meaningful components is a challenging inverse problem in image processing and has been widely applied to cartooning, texture removal, denoising, soft shadow/spotlight removal, detail enhancement etc. All the valuable contributions to this problem rely on a variational-based formulation where the intrinsic difficulty comes from the numerical intractability of the considered norms, from the tuning of the numerous model parameters, and, overall, from the complexity of extracting noise from a textured image, given the strong similarity between these two components. In this talk, I will present a two-stage variational model for the additive decomposition of images into piecewise constant, smooth, textured and white noise components, focusing on the regularization parameter selection and presenting numerical results of decomposition of textured images corrupted by several kinds of additive white noises.

Le 11 avril 2024 à 11:00

Séminaire Images Optimisation et Probabilités

salle de conférence

François Chapon Université of Toulouse

Spectres de matrices de Toeplitz à bande déformées

Les matrices de Toeplitz sont des matrices non-normales, dont l'analyse spectrale en grande dimension est bien comprise. Le spectre de ces matrices est en particulier très sensible à de petites perturbations. On s'intéressera dans cet exposé aux matrices de Toeplitz à bande, dont le symbole est donné par un polynôme de Laurent, et perturbées par une matrice aléatoire. Le but est de décrire les valeurs propres hors du support de la mesure limite de la perturbation quand la dimension tend vers l'infini, appelées "outliers", et qui apparaissent en fonction de l'indice de la courbe du plan complexe déterminée par le symbole. Travail en cours et en collaboration avec Mireille Capitaine et Charles Bordenave.

Le 18 avril 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférences

Pierre-Loïc Méliot Paris-Saclay

Ellis-Gärtner au second ordre pour des statistiques de tableaux et partitions aléatoires.

Si (X_n) est une suite de variables aléatoires réelles, le théorème d'Ellis-Gärtner assure que les logarithmes des probabilités de grandes déviations log P[X_n > n x] sont reliées à l'asymptotique de la log-laplace renormalisée h -> (log E[e^{h X_n}])/n. Dans cet exposé, on expliquera à quelles conditions on peut enlever les logarithmes et obtenir un équivalent des probabilités P[X_n > n x]. Plus précisément, si

log E[e^{z X_n}] = n Lambda(z) + Psi(z) + o(1)

localement uniformément sur le plan complexe, alors une condition simple sur la partie réelle de la fonction Lambda(z) permet d'écrire un équivalent des probabilités de grandes déviations (sans logarithme). Ces techniques s'adaptent en particulier à des modèles mettant en jeu des partitions aléatoires ou des tableaux de Young standards aléatoires ; nous détaillerons dans ce cas les résultats obtenus et quelques techniques de preuve générales.

Le 25 avril 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférénces

Maud Biquard ISAE-SUPAERO and CNES

Variational Bayes image restoration with (compressive) autoencoders

Regularization of inverse problems is of paramount importance in computational imaging. The ability of neural networks to learn efficient image representations has been recently exploited to design powerful data-driven regularizers. While state-of-the-art plug-and-play methods rely on an implicit regularization provided by neural denoisers, alternative Bayesian approaches consider Maximum A Posteriori (MAP) estimation in the latent space of a generative model, thus with an explicit regularization. However, state-of-the-art deep generative models require a huge amount of training data compared to denoisers. Besides, their complexity hampers the optimization involved in latent MAP derivation. In this work, we first propose to use compressive autoencoders instead. These networks, which can be seen as variational autoencoders with a flexible latent prior, are smaller and easier to train than state-of-the-art generative models. As a second contribution, we introduce the Variational Bayes Latent Estimation (VBLE) algorithm, which performs latent estimation within the framework of variational inference. Thanks to a simple yet efficient parameterization of the variational posterior, VBLE allows for fast and easy (approximate) posterior sampling. Experimental results on image datasets BSD and FFHQ demonstrate that VBLE reaches similar performance than state-of-the-art plug-and-play methods, while being able to quantify uncertainties faster than other existing posterior sampling techniques.

Le 2 mai 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférences

Jérôme Bolte Université de Toulouse 1

Nonsmooth differentiation of algorithms and solution maps

The recent surge in algorithmic differentiation through the massive use of TensorFlow and PyTorch "autodiff" has democratized "computerized differentiation" for a broad spectrum of applications and solvers. Motivated by the challenges of nonsmoothness (such as thresholding, constraints, and ReLU) and the need to adjust parameters in various contexts directly via these solvers, we have devised tools for nonsmooth differentiation compatible with autodiff. We have in particular developed a nonsmooth implicit function calculus, aiming to provide robust guarantees for prevalent differentiation practices. We will discuss applications of these findings through the differentiation of algorithms and equations.

Le 16 mai 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférénces

Bruno Galerne Université d'Orleans

À préciser

À préciser

Le 23 mai 2024 à 11:00

Séminaire Images Optimisation et Probabilités

Salle de conférénce

Stephane Dartois À préciser

À préciser

À préciser

Les anciens séminaires

Séminaire Images Optimisation et Probabilités

Institut de Mathématiques de Bordeaux UMR 5251