In this paper, we propose a procedure to accelerate the resolution of the well-known "Elastic-Net" problem. Our procedure is based on the (partial) identification of the solution support and the reformulation of the original problem into a problem of reduced dimension. The identification of the support leverages the novel concept of "safe relaxing" where one aims to identify non-zero coefficients of the solution. It can be viewed as a dual approach to "safe screening" introduced in the last decade and allowing to reduce the problem dimension using the identification of zero coefficients of the solution. We show numerically that combining both methodologies in a "Screen & Relax" strategy enables to significantly improve the trade-off between complexity and accuracy achievable by standard resolution techniques.

We present a novel screening methodology to safely discard irrelevant nodes within a generic branch-and-bound (BnB) algorithm solving the ℓ0-penalized least-squares problem. Our contribution is a set of two simple tests to detect sets of feasible vectors that cannot yield optimal solutions. This allows to prune nodes of the BnB search tree, thus reducing the overall optimization time. One cornerstone of our contribution is a nesting property between tests at different nodes that allows to implement them with a low computational cost. Our work leverages the concept of safe screening, well known for sparsity-inducing convex problems, and some recent advances in this field for ℓ0-penalized regression problems.

Acoustic echoes retrieval is a research topic that is gaining importance in many speech and audio signal processing applications such as speech enhancement, source separation, dereverberation and room geometry estimation. This work proposes a novel approach to blindly retrieve the off-grid timing of early acoustic echoes from a stereophonic recording of an unknown sound source such as speech. It builds on the recent framework of continuous dictionaries. In contrast with existing methods, the proposed approach does not rely on parameter tuning nor peak picking techniques by working directly in the parameter space of interest. The accuracy and robustness of the method are assessed on challenging simulated setups with varying noise and reverberation levels and are compared to two state-of-the-art methods.

Antisparse coding aims at spreading the information uniformly over representation coefficients and can be expressed as the solution of an ℓ∞-norm regularized problem. In this paper, we propose a new methodology, coined "safe squeezing", accelerating the computation of antisparse representations. The idea consists in identifying saturated entries of the solution via simple tests and compacting their contribution to achieve some form of dimensionality reduction. Numerical experiments show that the proposed approach leads to significant computational gain.

In this work, we present new theoretical results on sparse recovery guarantees for a greedy algorithm, orthogonal matching pursuit (OMP), in the context of continuous parametric dictionaries, i.e., made up of an infinite uncountable number of atoms. We build up a family of dictionaries for which k-step recovery is possible with OMP for 1-dimensional parameters. In higher dimension, algebraic conditions become necessary and will lead us to revisit some well-known k-step discrete analyses. Finally, a toy-example illustrates the level of tightness of our sufficient conditions.

Principal component analysis is a widely used technique to perform dimension reduction. However, selecting a finite number of significant components is essential and remains a crucial issue. Only few attempts have proposed a probabilistic approach to adaptively select this number. This paper introduces a Bayesian nonparametric model to jointly estimate the principal components and the corresponding intrinsic dimension. More precisely, the observations are projected onto a random orthogonal basis which is assigned a prior distribution defined on the Stiefel manifold. Then the factor scores take benefit of an Indian buffet process prior to model the uncertainty related to the number of components. The parameters of interest as well as the nuisance parameters are finally inferred within a fully Bayesian framework via Monte Carlo sampling. The performances of the proposed approach are assessed thanks to experiments conducted on various examples.