The behavior of the random feature model in the high-dimensional regression framework has become a popular issue of interest in the machine learning literature}. This model is generally considered for feature vectors $x_i = \Sigma^{1/2} x_i'$, where $x_i'$ is a random vector made of independent and identically distributed (iid) entries, and $\Sigma$ is a positive definite matrix representing the covariance of the features. In this paper, we move beyond {\CB this standard assumption by studying the performances of the random features model in the setting of non-iid feature vectors}. Our approach is related to the analysis of the spectrum of large random matrices through random matrix theory (RMT) {\CB and free probability} results. We turn to the analysis of non-iid data by using the notion of variance profile {\CB which} is {\CB well studied in RMT.} Our main contribution is then the study of the limits of the training and {\CB prediction} risks associated to the ridge estimator in the random features model when its dimensions grow. We provide asymptotic equivalents of these risks that capture the behavior of ridge regression with random features in a {\CB high-dimensional} framework. These asymptotic equivalents, {\CB which prove to be sharp in numerical experiments}, are retrieved by adapting, to our setting, established results from operator-valued free probability theory. Moreover, {\CB for various classes of random feature vectors that have not been considered so far in the literature}, our approach allows to show the appearance of the double descent phenomenon when the ridge regularization parameter is small enough.

This paper is focused on statistical learning from data that come as probability measures. In this setting, popular approaches consist in embedding such data into a Hilbert space with either Linearized Optimal Transport or Kernel Mean Embedding. However, the cost of computing such embeddings prohibits their direct use in large-scale settings. We study two methods based on measure quantization for approximating input probability measures with discrete measures of small-support size. The first one is based on optimal quantization of each input measure, while the second one relies on mean-measure quantization. We study the consistency of such approximations, and its implication for scalable embeddings of probability measures into a Hilbert space at a low computational cost. We finally illustrate our findings with various numerical experiments.

High-dimensional linear regression has been thoroughly studied in the context of independent and identically distributed data. We propose to investigate high-dimensional regression models for independent but non-identically distributed data. To this end, we suppose that the set of observed predictors (or features) is a random matrix with a variance profile and with dimensions growing at a proportional rate. Assuming a random effect model, we study the predictive risk of the ridge estimator for linear regression with such a variance profile. In this setting, we provide deterministic equivalents of this risk and of the degree of freedom of the ridge estimator. For certain class of variance profile, our work highlights the emergence of the well-known double descent phenomenon in high-dimensional regression for the minimum norm least-squares estimator when the ridge regularization parameter goes to zero. We also exhibit variance profiles for which the shape of this predictive risk differs from double descent. The proofs of our results are based on tools from random matrix theory in the presence of a variance profile that have not been considered so far to study regression models. Numerical experiments are provided to show the accuracy of the aforementioned deterministic equivalents on the computation of the predictive risk of ridge regression. We also investigate the similarities and differences that exist with the standard setting of independent and identically distributed data.

We introduce a new stochastic algorithm for solving entropic optimal transport (EOT) between two absolutely continuous probability measures $\mu$ and $\nu$. Our work is motivated by the specific setting of Monge-Kantorovich quantiles where the source measure $\mu$ is either the uniform distribution on the unit hypercube or the spherical uniform distribution. Using the knowledge of the source measure, we propose to parametrize a Kantorovich dual potential by its Fourier coefficients. In this way, each iteration of our stochastic algorithm reduces to two Fourier transforms that enables us to make use of the Fast Fourier Transform (FFT) in order to implement a fast numerical method to solve EOT. We study the almost sure convergence of our stochastic algorithm that takes its values in an infinite-dimensional Banach space. Then, using numerical experiments, we illustrate the performances of our approach on the computation of regularized Monge-Kantorovich quantiles. In particular, we investigate the potential benefits of entropic regularization for the smooth estimation of multivariate quantiles using data sampled from the target measure $\nu$.

This paper is focused on the study of entropic regularization in optimal transport as a smoothing method for Wasserstein estimators, through the prism of the classical tradeoff between approximation and estimation errors in statistics. Wasserstein estimators are defined as solutions of variational problems whose objective function involves the use of an optimal transport cost between probability measures. Such estimators can be regularized by replacing the optimal transport cost by its regularized version using an entropy penalty on the transport plan. The use of such a regularization has a potentially significant smoothing effect on the resulting estimators. In this work, we investigate its potential benefits on the approximation and estimation properties of regularized Wasserstein estimators. Our main contribution is to discuss how entropic regularization may reach, at a lowest computational cost, statistical performances that are comparable to those of un-regularized Wasserstein estimators in statistical learning problems involving distributional data analysis. To this end, we present new theoretical results on the convergence of regularized Wasserstein estimators. We also study their numerical performances using simulated and real data in the supervised learning problem of proportions estimation in mixture models using optimal transport.

This paper is concerned with the statistical analysis of matrix-valued time series. These are data collected over a network of sensors (typically a set of spatial locations), recording, over time, observations of multiple measurements. From such data, we propose to learn, in an online fashion, a graph that captures two aspects of dependency: one describing the sparse spatial relationship between sensors, and the other characterizing the measurement relationship. To this purpose, we introduce a novel multivariate autoregressive model to infer the graph topology encoded in the coefficient matrix which captures the sparse Granger causality dependency structure present in such matrix-valued time series. We decompose the graph by imposing a Kronecker sum structure on the coefficient matrix. We develop two online approaches to learn the graph in a recursive way. The first one uses Wald test for the projected OLS estimation, where we derive the asymptotic distribution for the estimator. For the second one, we formalize a Lasso-type optimization problem. We rely on homotopy algorithms to derive updating rules for estimating the coefficient matrix. Furthermore, we provide an adaptive tuning procedure for the regularization parameter. Numerical experiments using both synthetic and real data, are performed to support the effectiveness of the proposed learning approaches.