Stephane Chretien

Stephane Chretien, Nathalie Herr, Jean-Marc Nicod et al.

In a post-prognostics decision context, this paper addresses the problem of maximizing the useful life of a platform composed of several parallel machines under service constraint. Application on multi-stack fuel cell systems is considered. In order to propose a solution to the insufficient durability of fuel cells, the purpose is to define a commitment strategy by determining at each time the contribution of each fuel cell stack to the global output so as to satisfy the demand as long as possible. A relaxed version of the problem is introduced, which makes it potentially solvable for very large instances. Results based on computational experiments illustrate the efficiency of the new approach, based on the Mirror Prox algorithm, when compared with a simple method of successive projections onto the constraint sets associated with the problem.

Stephane Chretien

The goal of this note is to study the smallest conic singular value of a matrix from a Lagrangian duality viewpoint and provide an efficient method for its computation.

Emmanuel Caron, Stephane Chretien

Recent extensive numerical experiments in high scale machine learning have allowed to uncover a quite counterintuitive phase transition, as a function of the ratio between the sample size and the number of parameters in the model. As the number of parameters $p$ approaches the sample size $n$, the generalisation error increases, but surprisingly, it starts decreasing again past the threshold $p=n$. This phenomenon, brought to the theoretical community attention in \cite{belkin2019reconciling}, has been thoroughly investigated lately, more specifically for simpler models than deep neural networks, such as the linear model when the parameter is taken to be the minimum norm solution to the least-squares problem, firstly in the asymptotic regime when $p$ and $n$ tend to infinity, see e.g. \cite{hastie2019surprises}, and recently in the finite dimensional regime and more specifically for linear models \cite{bartlett2020benign}, \cite{tsigler2020benign}, \cite{lecue2022geometrical}. In the present paper, we propose a finite sample analysis of non-linear models of \textit{ridge} type, where we investigate the \textit{overparametrised regime} of the double descent phenomenon for both the \textit{estimation problem} and the \textit{prediction} problem. Our results provide a precise analysis of the distance of the best estimator from the true parameter as well as a generalisation bound which complements recent works of \cite{bartlett2020benign} and \cite{chinot2020benign}. Our analysis is based on tools closely related to the continuous Newton method \cite{neuberger2007continuous} and a refined quantitative analysis of the performance in prediction of the minimum $\ell_2$-norm solution.

Stephane Chretien, Zhen-Wai Olivier Ho

A problem of paramount importance in both pure (Restricted Invertibility problem) and applied mathematics (Feature extraction) is the one of selecting a submatrix of a given matrix, such that this submatrix has its smallest singular value above a specified level. Such problems can be addressed using perturbation analysis. In this paper, we propose a perturbation bound for the smallest singular value of a given matrix after appending a column, under the assumption that its initial coherence is not large, and we use this bound to derive a fast algorithm for feature extraction.

Stephane Chretien, Christophe Guyeux, Zhen-Wai Olivier HO

This paper studies the complexity of the stochastic gradient algorithm for PCA when the data are observed in a streaming setting. We also propose an online approach for selecting the learning rate. Simulation experiments confirm the practical relevance of the plain stochastic gradient approach and that drastic improvements can be achieved by learning the learning rate.

Stephane Chretien, Tianwen Wei

The subdifferential of convex functions of the singular spectrum of real matrices has been widely studied in matrix analysis, optimization and automatic control theory. Convex analysis and optimization over spaces of tensors is now gaining much interest due to its potential applications to signal processing, statistics and engineering. The goal of this paper is to present an applications to the problem of low rank tensor recovery based on linear random measurement by extending the results of Tropp to the tensors setting.