In the fist part of the talk, we will look to some statistical inverse problems for which the natural framework is no more an Euclidian one.
In the second part we will try to give the initial construction of (not orthogonal) wavelets -of the 80 - by Frazier, Jawerth,Weiss, before the Yves Meyer ORTHOGONAL wavelets theory.
In the third part we will propose a construction of a geometric wavelet theory. In the Euclidian case, Fourier transform plays a fundamental role. In the geometric situation this role is given to some "Laplacian operator" with some properties.
In the last part we will show that the previous theory could help to revisit the topic of regularity of Gaussian processes, and to give a criterium only based on the regularity of the covariance operator.[-]

A novel regularization technique based on random permutations of labels was proposed recently to tune hyper-parameters. Although this technique was successfully used in practice to train neural networks, little is known about its theoretical properties. This talk will present several results toward the theoretical understanding of this permutation technique in the simple Ridge regression model. These results combine a statistical analysis of the impact of random permutations on the geometry of the model with several novel perturbations bounds on the spectral decomposition of empirical covariance operators which can be of interest in themselves. Interestingly, this technique allows to build a novel smooth in-sample estimator of the excess risk which can be used to efficiently optimize the continuous hyper-parameters of regular models via gradient descent. This presentation is based in part on joint work with K. Meziani, G. Pacreau and B. Riu.[-]

We study the statistical behavior of empirical estimators of entropy-regularized optimal transport couplings between compact probability measures. These couplings were first proposed in a thought experiment of Schrödinger as a model for diffusing particles observed at different times, and have become popular in the last 10 years as computationally efficient proxies for optimal transport couplings. We review progress in characterizing rates of estimation for these estimators as well as their asymptotic limits. In particular, we describe a recent proof of a functional CLT conjectured by Harchaoui, Liu, and Pal (2020). Our proof is based on a stronger CLT for the dual solutions to the entropy-regularized problem in a suitable Hölder space. These CLTs also allow us to propose asymptotically valid goodness-of-fit tests based on the Sinkhorn divergence, a popular measure in machine learning. Based on joint work with E. del Barrio, A. González Sanz and J.-M. Loubes, and with G. Mena.[-]

We study the statistical behavior of empirical estimators of entropy-regularized optimal transport couplings between compact probability measures. These couplings were first proposed in a thought experiment of Schrödinger as a model for diffusing particles observed at different times, and have become popular in the last 10 years as computationally efficient proxies for optimal transport couplings. We review progress in characterizing rates of estimation for these estimators as well as their asymptotic limits. In particular, we describe a recent proof of a functional CLT conjectured by Harchaoui, Liu, and Pal (2020). Our proof is based on a stronger CLT for the dual solutions to the entropy-regularized problem in a suitable Hölder space. These CLTs also allow us to propose asymptotically valid goodness-of-fit tests based on the Sinkhorn divergence, a popular measure in machine learning. Based on joint work with E. del Barrio, A. González Sanz and J.-M. Loubes, and with G. Mena.[-]

It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints. We start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further study four concrete examples: a two-point testing problem, a potentially-diverging mean estimation problem, a nonparametric density estimation problem and a univariate median estimation problem. For each problem, we demonstrate procedures that are optimal in the presence of both contamination and LDP constraints, comment on the connections with the state-of-the-art methods that are only studied under either contamination or privacy constraints, and unveil the connections between robustness and LDP via partially answering whether LDP procedures are robust and whether robust procedures can be efficiently privatised. Overall, our work showcases a promising prospect of joint study for robustness and local differential privacy.
This is joint work with Mengchu Li and Yi Yu.[-]

The field of Robust Statistics studies the problem of designing estimators that perform well even when the data significantly deviates from the idealized modeling assumptions. The classical statistical theory, going back to the pioneering works by Tukey and Huber in the 1960s, characterizes the information-theoretic limits of robust estimation for a number of statistical tasks. On the other hand, until fairly recently, the computational aspects of this field were poorly understood. Specifically, no scalable robust estimation methods were known in high dimensions, even for the most basic task of mean estimation.
A recent line of work in computer science developed the first computationally efficient robust estimators in high dimensions for a range of learning tasks. This tutorial will provide an overview of these algorithmic developments and discuss some open problems in the area.[-]

The advent of large scale inference has spurred reexamination of conventional statistical thinking. In a series of highly original articles, Efron showed in some examples that the ensemble of the null distributed test statistics grossly deviated from the theoretical null distribution, and Efron persuasively illustrated the danger in assuming the theoretical null's veracity for downstream inference. Though intimidating in other contexts, the large scale setting is to the statistician's benefit here. There is now potential to estimate, rather than assume, the null distribution.
In a model for n many z-scores with at most k nonnulls, we adopt Efron's suggestion and consider estimation of location and scale parameters for a Gaussian null distribution. Placing no assumptions on the nonnull effects, we consider rate-optimal estimation in the entire regime k < n/2, that is, precisely the regime in which the null parameters are identifiable. The minimax upper bound is obtained by considering estimators based on the empirical characteristic function and the classical kernel mode estimator. Faster rates than those in Huber's contamination model are achievable by exploiting the Gaussian character of the data. As a consequence, it is shown that consistent estimation is indeed possible in the practically relevant regime k ≍ n. In a certain regime, the minimax lower bound involves constructing two marginal distributions whose characteristic functions match on a wide interval containing zero. The construction notably differs from those in the literature by sharply capturing a second-order scaling of n/2 − k in the minimax rate.[-]