arXiv subjects

Emmanuel Soubies

Publications and source records attributed to Emmanuel Soubies.

18 recordsLinked to original sources

arXiv2026

On The Linear Convergence of Bregman Proximal Gradient Methods with Applications to Kullback--Leibler regression

Bregman Proximal Gradient methods (BPGM) exploit the underlying geometry of the objective function through a carefully chosen mirror map. In this work, we introduce a novel notion of strong convexity, termed Restricted Relative Strong Convexity, and establish linear convergence rates for BPGM under this condition. We then exploit the proposed theoretical framework to provide an in-depth analysis of the convergence of BPGM for (regularized) Kullback--Leibler regression problems, covering scenarios with both unique and non-unique minimizers, as well as regularized and unregularized formulations. Specifically, we demonstrate that using the popular Burg's entropy as a distance-generating function may only yield linear convergence for certain KL regression problems. In contrast, we show that employing a smoothed version of the Burg's entropy induces the suitable geometry required to guarantee linear convergence. We conclude with numerical experiments that nicely align with our theoretical findings.

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arXiv2026

Relax and Follow: L0-Path Computation with L0-Bregman Relaxations

This work introduces L0PathBrex, a novel method for estimating the solution path of L0-regularized problems through the use of L0 Bregman relaxations (B-rex). Recently introduced and analyzed in the literature, these relaxations provide continuous reformulations of the original objective, are applicable to possibly non-quadratic data fidelity terms, and depend on a family of functions designed to preserve the global minimizers while eliminating part of the undesirable local minima. Given any numerical solver for the relaxation, the proposed approach dynamically constructs a collection of local minimizers that are candidates for the L0-solution path. It exploits warm-start strategies and identifies ranges of the regularization parameter for which each minimizer remains valid under the corresponding relaxation. Experiments on sparse least-squares and logistic regression problems demonstrate that L0PathBrex systematically outperforms state-of-the-art baselines across both synthetic and real-world datasets in terms of various evaluation metrics; additionally, the study investigates how the choice of the B-rex affects the quality of the estimated path in the sparse Poisson regression setting.

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arXiv2025

Optimization landscape of $\ell_0$-Bregman relaxations

In this paper, we study (noisy) linear systems, and their $\ell_0$-regularized optimization problems, coupled with general data fidelity terms. Recent approaches for solving this class of problems have proposed to consider non-convex exact continuous relaxations that preserve global minimizers while reducing the number of local minimizers. Within this framework, we consider the class of $\ell_0$-Bregman relaxations, and establish sufficient conditions under which a critical point is isolated in terms of sparsity, in the sense that any other critical point has a strictly larger cardinality. In this way, we ensure a form of uniqueness in the solution structure. Furthermore, we analyze the exact recovery properties of such exact relaxations. To that end, we derive conditions under which the oracle solution (i.e., the one sharing the same support as the ground-truth) is the unique global minimizer of the relaxed problem, and is isolated in terms of sparsity. Our analysis is primarily built upon a novel property we introduce, termed the Bregman Restricted Strong Convexity. Finally, we specialize our general results to both sparse Gaussian (least-squares) and Poisson ((generalized) Kullback-Leibler divergence) regression problems. In particular, we show that our general analysis sharpens existing bounds for the LS setting, while providing an entirely new result for the KL case.

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arXiv2025

Proximal Operators of Sorted Nonconvex Penalties

This work studies the problem of sparse signal recovery with automatic grouping of variables. To this end, we investigate sorted nonsmooth penalties as a regularization approach for generalized linear models. We focus on a family of sorted nonconvex penalties which generalizes the Sorted L1 Norm (SLOPE). These penalties are designed to promote clustering of variables due to their sorted nature, while the nonconvexity reduces the shrinkage of coefficients. Our goal is to provide efficient ways to compute their proximal operator, enabling the use of popular proximal algorithms to solve composite optimization problems with this choice of sorted penalties. We distinguish between two classes of problems: the weakly convex case where computing the proximal operator remains a convex problem, and the nonconvex case where computing the proximal operator becomes a challenging nonconvex combinatorial problem. For the weakly convex case (e.g. sorted MCP and SCAD), we explain how the Pool Adjacent Violators (PAV) algorithm can exactly compute the proximal operator. For the nonconvex case (e.g. sorted Lq with q in ]0,1[), we show that a slight modification of this algorithm turns out to be remarkably efficient to tackle the computation of the proximal operator. We also present new theoretical insights on the minimizers of the nonconvex proximal problem. We demonstrate the practical interest of using such penalties on several experiments.

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arXiv2025

Convexity in ReLU Neural Networks: beyond ICNNs?

Convex functions and their gradients play a critical role in mathematical imaging, from proximal optimization to Optimal Transport. The successes of deep learning has led many to use learning-based methods, where fixed functions or operators are replaced by learned neural networks. Regardless of their empirical superiority, establishing rigorous guarantees for these methods often requires to impose structural constraints on neural architectures, in particular convexity. The most popular way to do so is to use so-called Input Convex Neural Networks (ICNNs). In order to explore the expressivity of ICNNs, we provide necessary and sufficient conditions for a ReLU neural network to be convex. Such characterizations are based on product of weights and activations, and write nicely for any architecture in the path-lifting framework. As particular applications, we study our characterizations in depth for 1 and 2-hidden-layer neural networks: we show that every convex function implemented by a 1-hidden-layer ReLU network can be also expressed by an ICNN with the same architecture; however this property no longer holds with more layers. Finally, we provide a numerical procedure that allows an exact check of convexity for ReLU neural networks with a large number of affine regions.

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arXiv2025

Box-constrained L0 Bregman-relaxations

Regularization using the L0 pseudo-norm is a common approach to promote sparsity, with widespread applications in machine learning and signal processing. However, solving such problems is known to be NP-hard. Recently, the L0 Bregman relaxation (B-rex) has been introduced as a continuous, non-convex approximation of the L0 pseudo-norm. Replacing the L0 term with B-rex leads to exact continuous relaxations that preserve the global optimum while simplifying the optimization landscape, making non-convex problems more tractable for algorithmic approaches. In this paper, we focus on box-constrained exact continuous Bregman relaxations of L0-regularized criteria with general data terms, including least-squares, logistic regression, and Kullback-Leibler fidelities. Experimental results on synthetic data, compared with Branch-and-Bound methods, demonstrate the effectiveness of the proposed relaxations.

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arXiv2024

Exact continuous relaxations of l0-regularized criteria with non-quadratic data terms

We propose a new class of exact continuous relaxations of l0-regularized criteria involving non-quadratic data terms such as the Kullback-Leibler divergence and the logistic regression, possibly combined with an l2 regularization. We first prove the existence of global minimizers for such problems and characterize their local minimizers.Then, we propose the l0 Bregman Relaxation (B-rex), a continuous approximation of the l0 pseudo-norm defined in terms of suitable Bregman distances, which leads to an exact continuous relaxations of the original l0-regularized problem in the sense that it does not alter its set of global minimizers and reduces the non-convexity by eliminating certain local minimizers. Both features make the relaxed problem more amenable to be solved by standard non-convex optimization algorithms. In this spirit, we consider the proximal gradient algorithm and provide explicit computation of proximal points for the B-rex penalty in several cases. Finally, we report a set of numerical results illustrating the geometrical behavior of the proposed B-rex penalty for different choices of the underlying Bregman distance, its relation with convex envelopes, as well as its exact relaxation properties in 1D/2D and higher dimensions.

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arXiv2024

Automatic Tuning of Denoising Algorithms Parameters Without Ground Truth

Denoising is omnipresent in image processing. It is usually addressed with algorithms relying on a set of hyperparameters that control the quality of the recovered image. Manual tuning of those parameters can be a daunting task, which calls for the development of automatic tuning methods. Given a denoising algorithm, the best set of parameters is the one that minimizes the error between denoised and ground-truth images. Clearly, this ideal approach is unrealistic, as the ground-truth images are unknown in practice. In this work, we propose unsupervised cost functions -- i.e., that only require the noisy image -- that allow us to reach this ideal gold standard performance. Specifically, the proposed approach makes it possible to obtain an average PSNR output within less than 1% of the best achievable PSNR.

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arXiv2023

Optical Diffraction Tomography Meets Fluorescence Localization Microscopy

We show that structural information can be extracted from single molecule localization microscopy (SMLM) data. More precisely, we reinterpret SMLM data as the measures of a phaseless optical diffraction tomography system for which the illumination sources are fluorophores within the sample. Building upon this model, we propose a joint optimization framework to estimate both the refractive index map and the position of fluorescent molecules from the sole SMLM frames.

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arXiv2023

Deep Unrolling for Nonconvex Robust Principal Component Analysis

We design algorithms for Robust Principal Component Analysis (RPCA) which consists in decomposing a matrix into the sum of a low rank matrix and a sparse matrix. We propose a deep unrolled algorithm based on an accelerated alternating projection algorithm which aims to solve RPCA in its nonconvex form. The proposed procedure combines benefits of deep neural networks and the interpretability of the original algorithm and it automatically learns hyperparameters. We demonstrate the unrolled algorithm's effectiveness on synthetic datasets and also on a face modeling problem, where it leads to both better numerical and visual performances.

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arXiv2023

Accelerating Non-Negative and Bounded-Variable Linear Regression Algorithms with Safe Screening

Non-negative and bounded-variable linear regression problems arise in a variety of applications in machine learning and signal processing. In this paper, we propose a technique to accelerate existing solvers for these problems by identifying saturated coordinates in the course of iterations. This is akin to safe screening techniques previously proposed for sparsity-regularized regression problems. The proposed strategy is provably safe as it provides theoretical guarantees that the identified coordinates are indeed saturated in the optimal solution. Experimental results on synthetic and real data show compelling accelerations for both non-negative and bounded-variable problems.

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arXiv2023

Expanding boundaries of Gap Safe screening

Sparse optimization problems are ubiquitous in many fields such as statistics, signal/image processing and machine learning. This has led to the birth of many iterative algorithms to solve them. A powerful strategy to boost the performance of these algorithms is known as safe screening: it allows the early identification of zero coordinates in the solution, which can then be eliminated to reduce the problem's size and accelerate convergence. In this work, we extend the existing Gap Safe screening framework by relaxing the global strong-concavity assumption on the dual cost function. Instead, we exploit local regularity properties, that is, strong concavity on well-chosen subsets of the domain. The non-negativity constraint is also integrated to the existing framework. Besides making safe screening possible to a broader class of functions that includes beta-divergences (e.g., the Kullback-Leibler divergence), the proposed approach also improves upon the existing Gap Safe screening rules on previously applicable cases (e.g., logistic regression). The proposed general framework is exemplified by some notable particular cases: logistic function, beta = 1.5 and Kullback-Leibler divergences. Finally, we showcase the effectiveness of the proposed screening rules with different solvers (coordinate descent, multiplicative-update and proximal gradient algorithms) and different data sets (binary classification, hyperspectral and count data).

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arXiv2022

Leveraging Joint-Diagonalization in Transform-Learning NMF

Non-negative matrix factorization with transform learning (TL-NMF) is a recent idea that aims at learning data representations suited to NMF. In this work, we relate TL-NMF to the classical matrix joint-diagonalization (JD) problem. We show that, when the number of data realizations is sufficiently large, TL-NMF can be replaced by a two-step approach -- termed as JD+NMF -- that estimates the transform through JD, prior to NMF computation. In contrast, we found that when the number of data realizations is limited, not only is JD+NMF no longer equivalent to TL-NMF, but the inherent low-rank constraint of TL-NMF turns out to be an essential ingredient to learn meaningful transforms for NMF.

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arXiv2020

Joint Angular Refinement and Reconstruction for Single-Particle Cryo-EM

Single-particle cryo-electron microscopy (cryo-EM) reconstructs the three-dimensional (3D) structure of bio-molecules from a large set of 2D projection images with random and unknown orientations. A crucial step in the single-particle cryo-EM pipeline is 3D refinement, which resolves a high-resolution 3D structure from an initial approximate volume by refining the estimation of the orientation of each projection. In this work, we propose a new approach that refines the projection angles on the continuum. We formulate the optimization problem over the density map and the orientations jointly. The density map is updated using the efficient alternating-direction method of multipliers, while the orientations are updated through a semi-coordinate-wise gradient descent for which we provide an explicit derivation of the gradient. Our method eliminates the requirement for a fine discretization of the orientation space and does away with the classical but computationally expensive template-matching step. Numerical results demonstrate the feasibility and performance of our approach compared to several baselines.

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arXiv2019

Three-Dimensional Optical Diffraction Tomography with Lippmann-Schwinger Model

A broad class of imaging modalities involve the resolution of an inverse-scattering problem. Among them, three-dimensional optical diffraction tomography (ODT) comes with its own challenges. These include a limited range of views, a large size of the sample with respect to the illumination wavelength, and optical aberrations that are inherent to the system itself. In this work, we present an accurate and efficient implementation of the forward model. It relies on the exact (nonlinear) Lippmann-Schwinger equation. We address several crucial issues such as the discretization of the Green function, the computation of the far field, and the estimation of the incident field. We then deploy this model in a regularized variational-reconstruction framework and show on both simulated and real data that it leads to substantially better reconstructions than the approximate models that are traditionally used in ODT.

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arXiv2019

Pocket Guide to Solve Inverse Problems with GlobalBioIm

GlobalBioIm is an open-source MATLAB library for solving inverse problems. The library capitalizes on the strong commonalities between forward models to standardize the resolution of a wide range of imaging inverse problems. Endowed with an operator-algebra mechanism, GlobalBioIm allows one to easily solve inverse problems by combining elementary modules in a lego-like fashion. This user-friendly toolbox gives access to cutting-edge reconstruction algorithms, while its high modularity makes it easily extensible to new modalities and novel reconstruction methods. We expect GlobalBioIm to respond to the needs of imaging scientists looking for reliable and easy-to-use computational tools for solving their inverse problems. In this paper, we present in detail the structure and main features of the library. We also illustrate its flexibility with examples from multichannel deconvolution microscopy.

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arXiv2018

The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy

This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem. The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.

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arXiv2017

Efficient Inversion of Multiple-Scattering Model for Optical Diffraction Tomography

Optical diffraction tomography relies on solving an inverse scattering problem governed by the wave equation. Classical reconstruction algorithms are based on linear approximations of the forward model (Born or Rytov), which limits their applicability to thin samples with low refractive-index contrasts. More recent works have shown the benefit of adopting nonlinear models. They account for multiple scattering and reflections, improving the quality of reconstruction. To reduce the complexity and memory requirements of these methods, we derive an explicit formula for the Jacobian matrix of the nonlinear Lippmann-Schwinger model which lends itself to an efficient evaluation of the gradient of the data- fidelity term. This allows us to deploy efficient methods to solve the corresponding inverse problem subject to sparsity constraints.

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