Séminaire Images Optimisation et Probabilités
Responsable : Luis Fredes et Camille Male
Travail en collaboration avec Y. Gueddari et W. Hachem, voir aussi https://arxiv.org/abs/2302.07820
Based on joint work with Yurii Nesterov.
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.
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.
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.
Neural style transfer (NST) is a deep learning technique that produces an unprecedentedly rich style transfer from a style image to a content image. It is particularly impressive when it comes to transferring style from a painting to an image. NST was originally achieved by solving an optimization problem to match the global statistics of the style image while preserving the local geometric features of the content image. The two main drawbacks of this original approach is that it is computationally expensive and that the resolution of the output images is limited by high GPU memory requirements. Many solutions have been proposed to both accelerate NST and produce images with larger size. However, our investigation shows that these accelerated methods all compromise the quality of the produced images in the context of painting style transfer. Indeed, transferring the style of a painting is a complex task involving features at different scales, from the color palette and compositional style to the fine brushstrokes and texture of the canvas. This paper provides a solution to solve the original global optimization for ultra-high resolution (UHR) images, enabling multiscale NST at unprecedented image sizes. This is achieved by spatially localizing the computation of each forward and backward passes through the VGG network. Extensive qualitative and quantitative comparisons, as well as a user study, show that our method produces style transfer of unmatched quality for such high-resolution painting styles. By a careful comparison, we show that state of the art fast methods are still prone to artifacts, thus suggesting that fast painting style transfer remains an open problem.
Joint work with Lara Raad, José Lezama and Jean-Michel Morel.
In this talk, I will present the results of a collaboration with Benjamin McKenna on the injective norm of large random Gaussian tensors and uniform random quantum states and, time allowing, describe some of the context underlying this work. The injective norm is a natural generalization to tensors of the operator norm of a matrix and appears in multiple fields. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, known as geometric entanglement. In our recent preprint, we provide high-probability upper bounds in different regimes on the injective norm of real and complex Gaussian random tensors, which corresponds to lower bounds on the geometric entanglement of random quantum states, and to bounds on the ground-state energy of a particular multispecies spherical spin glass model. Our result represents a first step towards solving an important question in quantum information that has been part of folklore.
Optimal Transport is a useful metric to compare probability distributions and to compute a pairing given a ground cost. Its entropic regularization variant (eOT) is crucial to have fast algorithms and reflect fuzzy/noisy matchings. This work focuses on Inverse Optimal Transport (iOT), the problem of inferring the ground cost from samples drawn from a coupling that solves an eOT problem. It is a relevant problem that can be used to infer unobserved/missing links, and to obtain meaningful information about the structure of the ground cost yielding the pairing. On one side, iOT benefits from convexity, but on the other side, being ill-posed, it requires regularization to handle the sampling noise. This work presents an in-depth theoretical study of the l1 regularization to model for instance Euclidean costs with sparse interactions between features. Specifically, we derive a sufficient condition for the robust recovery of the sparsity of the ground cost that can be seen as a far reaching generalization of the Lasso's celebrated Irrepresentability Condition. To provide additional insight into this condition, we work out in detail the Gaussian case. We show that as the entropic penalty varies, the iOT problem interpolates between a graphical Lasso and a classical Lasso, thereby stablishing a connection between iOT and graph estimation, an important problem in ML.
Page de l'événement : https://indico.math.cnrs.fr/event/11353/overview
Mercredi 10/07
14h00 Jurgen Angst (Univ. Rennes)
Title : TLC in total variation for beta-ensembles
Abstract : In this talk, we study the fluctuations of linear statistics associated with beta-ensembles, which are statistical physics models generalizing random matrix spectra. In the context of random matrices precisely (e.g. GOE, GUE), the "law of large numbers" is Wigner's theorem, which states that the empirical measure of eigenvalues converges to the semicircle law, and fluctuations around equilibrium can be shown to be Gaussian. We will describe how this result generalizes to beta-ensembles and how it is possible to quantify the speed of convergence to the normal distribution. We obtain optimal rates of convergence for the total variation distance and the Wasserstein distances. To do this, we introduce a variant of Stein's method for a generator $L$ that is not necessarily invertible, and which allows us to establish the asymptotic normality of observables that are not in the image of $L$. Time permitting, we will also look at the phenomenon of super-convergence, which ensures that convergence to the normal law takes place for very strong metrics, typically the $C^{\infty}$-convergence of densities. The talk is based on recent works with R. Herry, D. Malicet and G. Poly.
15h00 Nicolas Juillet (Univ. Haute-Alsace)
Title : Exact interpolation of 1-marginals
Abstract : I shall present a new type of martingales that exactly interpolates any given family of 1-dimensional marginals on R1 (satisfying the suitable necessary assumption). The construction makes use of ideas from the (martingale) optimal transportation theory and relies on different stochastic orders. I shall discuss of related constructions and open questions (joint work with Brückerhoff and Huesmann).
16h00 Kolehe Coulibaly-Pasquier (Inst. Ellie Cartan)
Title : On the separation cut-off phenomenon for Brownian motions on high dimensional rotationally
symmetric compact manifolds.
Abstract : Given a family of compact, rotationally symmetric manifolds indexed by the dimension and a weighted function, we will study the cut-off phenomena for the Brownian motion on this family.
Our proof is based on the construction of an intertwined process, a strong stationary time, an estimation of the moments of the covering time of the dual process, and on the phenomena of concentration of the measure.
We will see a phase transition concerning the existence or not of cut-off phenomena, which depend on the shape of the weighted function.
Jeudi 11/07
11h00 Masha Gordina (Univ. of Connecticut)
Title : Dimension-independent functional inequalities on sub-Riemannian manifolds
Abstract : The talk will review recent results on gradient estimates, log Sobolev inequalities, reverse Poincare and log Sobolev inequalities on a class of sub-Riemannian manifolds. As for many of such setting curvature bounds are not available, we use different techniques including tensorization and taking quotients. Joint work with F. Baudoin, L. Luo and R. Sarkar.
À preciser
Institut de Mathématiques de Bordeaux UMR 5251
Université de Bordeaux
351, cours de la Libération - F 33 405 TALENCE
Tél: +33 (0)5 40 00 60 70
institut
math.u-bordeaux.fr
