arXiv subjects

Clément Elvira

Publications and source records attributed to Clément Elvira.

15 recordsLinked to original sources

arXiv2026

Physically constrained unfolded multi-dimensional OMP for large MIMO systems

Sparse recovery methods are essential for channel estimation and localization in modern communication systems, but their reliability relies on accurate physical models, which are rarely perfectly known. Their computational complexity also grows rapidly with the dictionary dimensions in large MIMO systems. In this paper, we propose MOMPnet, a novel unfolded sparse recovery framework that addresses both the reliability and complexity challenges of traditional methods. By integrating deep unfolding with data-driven dictionary learning, MOMPnet mitigates hardware impairments while preserving interpretability. Instead of a single large dictionary, multiple smaller, independent dictionaries are employed, enabling a low-complexity multidimensional Orthogonal Matching Pursuit algorithm. The proposed unfolded network is evaluated on realistic channel data against multiple baselines, demonstrating its strong performance and potential.

eess.SP↗

arXiv2025

Model-based learning for joint channel estimationand hybrid MIMO precoding

Hybrid precoding is a key ingredient of cost-effective massive multiple-input multiple-output transceivers. However, setting jointly digital and analog precoders to optimally serve multiple users is a difficult optimization problem. Moreover, it relies heavily on precise knowledge of the channels, which is difficult to obtain, especially when considering realistic systems comprising hardware impairments. In this paper, a joint channel estimation and hybrid precoding method is proposed, which consists in an end-to-end architecture taking received pilots as inputs and outputting pre-coders. The resulting neural network is fully model-based, making it lightweight and interpretable with very few learnable parameters. The channel estimation step is performed using the unfolded matching pursuit algorithm, accounting for imperfect knowledge of the antenna system, while the precoding step is done via unfolded projected gradient ascent. The great potential of the proposed method is empirically demonstrated on realistic synthetic channels.

eess.SP↗

arXiv2025

A Generic Branch-and-Bound Algorithm for $\ell_0$-Penalized Problems with Supplementary Material

We present a generic Branch-and-Bound procedure designed to solve L0-penalized optimization problems. Existing approaches primarily focus on quadratic losses and construct relaxations using "Big-M" constraints and/or L2-norm penalties. In contrast, our method accommodates a broader class of loss functions and allows greater flexibility in relaxation design through a general penalty term, encompassing existing techniques as special cases. We establish theoretical results ensuring that all key quantities required for the Branch-and-Bound implementation admit closed-form expressions under the general blanket assumptions considered in our work. Leveraging this framework, we introduce El0ps, an open-source Python solver with a plug-and-play workflow that enables user-defined losses and penalties in L0-penalized problems. Through extensive numerical experiments, we demonstrate that El0ps achieves state-of-the-art performance on classical instances and extends computational feasibility to previously intractable ones.

math.OC↗

arXiv2025

El0ps: An Exact L0-regularized Problems Solver

This paper presents El0ps, a Python toolbox providing several utilities to handle L0-regularized problems related to applications in machine learning, statistics, and signal processing, among other fields. In contrast to existing toolboxes, El0ps allows users to define custom instances of these problems through a flexible framework, provides a dedicated solver achieving state-of-the-art performance, and offers several built-in machine learning pipelines. Our aim with El0ps is to provide a comprehensive tool which opens new perspectives for the integration of L0-regularized problems in practical applications.

cs.MS↗

arXiv2024

One to beat them all: "RYU" -- a unifying framework for the construction of safe balls

In this paper, we present a new framework, called "RYU" for constructing "safe" regions -- specifically, bounded sets that are guaranteed to contain the dual solution of a target optimization problem. Our framework applies to the standard case where the objective function is composed of two components: a closed, proper, convex function with Lipschitz-smooth gradient and another closed, proper, convex function. We show that the RYU framework not only encompasses but also improves upon the state-of-the-art methods proposed over the past decade for this class of optimization problems.

math.OC↗

arXiv2024

A New Branch-and-Bound Pruning Framework for $\ell_0$-Regularized Problems

We consider the resolution of learning problems involving $\ell_0$-regularization via Branch-and-Bound (BnB) algorithms. These methods explore regions of the feasible space of the problem and check whether they do not contain solutions through "pruning tests". In standard implementations, evaluating a pruning test requires to solve a convex optimization problem, which may result in computational bottlenecks. In this paper, we present an alternative to implement pruning tests for some generic family of $\ell_0$-regularized problems. Our proposed procedure allows the simultaneous assessment of several regions and can be embedded in standard BnB implementations with a negligible computational overhead. We show through numerical simulations that our pruning strategy can improve the solving time of BnB procedures by several orders of magnitude for typical problems encountered in machine-learning applications.

math.OC↗

arXiv2023

Safe Peeling for L0-Regularized Least-Squares with supplementary material

We introduce a new methodology dubbed ``safe peeling'' to accelerate the resolution of L0-regularized least-squares problems via a Branch-and-Bound (BnB) algorithm. Our procedure enables to tighten the convex relaxation considered at each node of the BnB decision tree and therefore potentially allows for more aggressive pruning. Numerical simulations show that our proposed methodology leads to significant gains in terms of number of nodes explored and overall solving time.s show that our proposed methodology leads to significant gains in terms of number of nodes explored and overall solving time.

cs.LG↗

arXiv2022

Safe rules for the identification of zeros in the solutions of the SLOPE problem

In this paper we propose a methodology to accelerate the resolution of the so-called "Sorted L-One Penalized Estimation" (SLOPE) problem. Our method leverages the concept of "safe screening", well-studied in the literature for \textit{group-separable} sparsity-inducing norms, and aims at identifying the zeros in the solution of SLOPE. More specifically, we derive a set of $\tfrac{n(n+1)}{2}$ inequalities for each element of the $n$-dimensional primal vector and prove that the latter can be safely screened if some subsets of these inequalities are verified. We propose moreover an efficient algorithm to jointly apply the proposed procedure to all the primal variables. Our procedure has a complexity $\mathcal{O}(n\log n + LT)$ where $T\leq n$ is a problem-dependent constant and $L$ is the number of zeros identified by the tests. Numerical experiments confirm that, for a prescribed computational budget, the proposed methodology leads to significant improvements of the solving precision.

cs.LG↗

arXiv2022

Beyond GAP screening for Lasso by exploiting new dual cutting half-spaces with supplementary material

In this paper, we propose a novel safe screening test for Lasso. Our procedure is based on a safe region with a dome geometry and exploits a canonical representation of the set of half-spaces (referred to as "dual cutting half-spaces" in this paper) containing the dual feasible set. The proposed safe region is shown to be always included in the state-of-the-art "GAP Sphere" and "GAP Dome" proposed by Fercoq et al. (and strictly so under very mild conditions) while involving the same computational burden. Numerical experiments confirm that our new dome enables to devise more powerful screening tests than GAP regions and lead to significant acceleration to solve Lasso.

cs.LG↗

arXiv2022

Screen & Relax: Accelerating the resolution of Elastic-net by safe identification of the solution support

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 tradeoff between complexity and accuracy achievable by standard resolution techniques.

math.OC↗

arXiv2022

Node-screening tests for L0-penalized least-squares problem with supplementary material

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.

eess.SP↗

arXiv2020

When does OMP achieve exact recovery with continuous dictionaries?

This paper presents new theoretical results on sparse recovery guarantees for a greedy algorithm, Orthogonal Matching Pursuit (OMP), in the context of continuous parametric dictionaries. Here, the continuous setting means that the dictionary is made up of an infinite uncountable number of atoms. In this work, we rely on the Hilbert structure of the observation space to express our recovery results as a property of the kernel defined by the inner product between two atoms. Using a continuous extension of Tropp's Exact Recovery Condition, we identify key assumptions allowing to analyze OMP in the continuous setting. Under these assumptions, OMP unambiguously identifies in exactly $k$ steps the atom parameters from any observed linear combination of $k$ atoms. These parameters play the role of the so-called support of a sparse representation in traditional sparse recovery. In our paper, any kernel and set of parameters that satisfy these conditions are said to be admissible. In the one-dimensional setting, we exhibit a family of kernels relying on completely monotone functions for which admissibility holds for any set of atom parameters. For higher dimensional parameter spaces, the analysis turns out to be more subtle. An additional assumption, so-called axis admissibility, is imposed to ensure a form of delayed recovery (in at most $k^D$ steps, where $D$ is the dimension of the parameter space). Furthermore, guarantees for recovery in exactly $k$ steps are derived under an additional algebraic condition involving a finite subset of atoms (built as an extension of the set of atoms to be recovered). We show that the latter technical conditions simplify in the case of Laplacian kernels, allowing us to derive simple conditions for $k$-step exact recovery, and to carry out a coherence-based analysis in terms of a minimum separation assumption between the atoms to be recovered.

cs.IT↗

arXiv2020

Safe squeezing for antisparse coding

Spreading the information over all coefficients of a representation is a desirable property in many applications such as digital communication or machine learning. This so-called antisparse representation can be obtained by solving a convex program involving an $\ell_\infty$-norm penalty combined with a quadratic discrepancy. In this paper, we propose a new methodology, dubbed safe squeezing, to accelerate the computation of antisparse representation. We describe a test that allows to detect saturated entries in the solution of the optimization problem. The contribution of these entries is compacted into a single vector, thus operating a form of dimensionality reduction. We propose two algorithms to solve the resulting lower dimensional problem. Numerical experiments show the effectiveness of the proposed method to detect the saturated components of the solution and illustrates the induced computational gains in the resolution of the antisparse problem.

cs.LG↗

arXiv2017

Bayesian nonparametric Principal Component Analysis

Principal component analysis (PCA) is very popular to perform dimension reduction. The selection of the number of significant components is essential but often based on some practical heuristics depending on the application. Only few works have proposed a probabilistic approach able to infer the number of significant components. To this purpose, this paper introduces a Bayesian nonparametric principal component analysis (BNP-PCA). The proposed model projects observations onto a random orthogonal basis which is assigned a prior distribution defined on the Stiefel manifold. The prior on factor scores involves an Indian buffet process 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. A study of the (in-)consistence of the marginal maximum a posteriori estimator of the latent dimension is carried out. A new estimator of the subspace dimension is proposed. Moreover, for sake of statistical significance, a Kolmogorov-Smirnov test based on the posterior distribution of the principal components is used to refine this estimate. The behaviour of the algorithm is first studied on various synthetic examples. Finally, the proposed BNP dimension reduction approach is shown to be easily yet efficiently coupled with clustering or latent factor models within a unique framework.

stat.ML↗

arXiv2015

Bayesian anti-sparse coding

Sparse representations have proven their efficiency in solving a wide class of inverse problems encountered in signal and image processing. Conversely, enforcing the information to be spread uniformly over representation coefficients exhibits relevant properties in various applications such as digital communications. Anti-sparse regularization can be naturally expressed through an $\ell_{\infty}$-norm penalty. This paper derives a probabilistic formulation of such representations. A new probability distribution, referred to as the democratic prior, is first introduced. Its main properties as well as three random variate generators for this distribution are derived. Then this probability distribution is used as a prior to promote anti-sparsity in a Gaussian linear inverse problem, yielding a fully Bayesian formulation of anti-sparse coding. Two Markov chain Monte Carlo (MCMC) algorithms are proposed to generate samples according to the posterior distribution. The first one is a standard Gibbs sampler. The second one uses Metropolis-Hastings moves that exploit the proximity mapping of the log-posterior distribution. These samples are used to approximate maximum a posteriori and minimum mean square error estimators of both parameters and hyperparameters. Simulations on synthetic data illustrate the performances of the two proposed samplers, for both complete and over-complete dictionaries. All results are compared to the recent deterministic variational FITRA algorithm.

stat.ML↗