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Mathurin Massias

Publications and source records attributed to Mathurin Massias.

At least 19 recordsLinked to original sources
arXiv2026

Training Flow Matching: The Role of Weighting and Parameterization

We study the training objectives of denoising-based generative models, with a particular focus on loss weighting and output parameterization, including noise-, clean image-, and velocity-based formulations. Through a systematic numerical study, we analyze how these training choices interact with the intrinsic dimensionality of the data manifold, model architecture, and dataset size. Our experiments span synthetic datasets with controlled geometry as well as image data, and compare training objectives using quantitative metrics for denoising accuracy (PSNR across noise levels) and generative quality (FID). Rather than proposing a new method, our goal is to disentangle the various factors that matter when training a flow matching model, in order to provide practical insights on design choices.

cs.CV↗
arXiv2025

Efficient Solvers for SLOPE in R, Python, Julia, and C++

We present a suite of packages in R, Python, Julia, and C++ that efficiently solve the Sorted L-One Penalized Estimation (SLOPE) problem. The packages feature a highly efficient hybrid coordinate descent algorithm that fits generalized linear models (GLMs) and supports a variety of loss functions, including Gaussian, binomial, Poisson, and multinomial logistic regression. Our implementation is designed to be fast, memory-efficient, and flexible. The packages support a variety of data structures (dense, sparse, and out-of-memory matrices) and are designed to efficiently fit the full SLOPE path as well as handle cross-validation of SLOPE models, including the relaxed SLOPE. We present examples of how to use the packages and benchmarks that demonstrate the performance of the packages on both real and simulated data and show that our packages outperform existing implementations of SLOPE in terms of speed.

stat.CO↗
arXiv2025

The Generation Phases of Flow Matching: a Denoising Perspective

Flow matching has achieved remarkable success, yet the factors influencing the quality of its generation process remain poorly understood. In this work, we adopt a denoising perspective and design a framework to empirically probe the generation process. Laying down the formal connections between flow matching models and denoisers, we provide a common ground to compare their performances on generation and denoising. This enables the design of principled and controlled perturbations to influence sample generation: noise and drift. This leads to new insights on the distinct dynamical phases of the generative process, enabling us to precisely characterize at which stage of the generative process denoisers succeed or fail and why this matters.

cs.CV↗
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.

math.OC↗
arXiv2025

On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity

Modern deep generative models can now produce high-quality synthetic samples that are often indistinguishable from real training data. A growing body of research aims to understand why recent methods, such as diffusion and flow matching techniques, generalize so effectively. Among the proposed explanations are the inductive biases of deep learning architectures and the stochastic nature of the conditional flow matching loss. In this work, we rule out the noisy nature of the loss as a key factor driving generalization in flow matching. First, we empirically show that in high-dimensional settings, the stochastic and closed-form versions of the flow matching loss yield nearly equivalent losses. Then, using state-of-the-art flow matching models on standard image datasets, we demonstrate that both variants achieve comparable statistical performance, with the surprising observation that using the closed-form can even improve performance.

cs.LG↗
arXiv2025

DeepInverse: A Python package for solving imaging inverse problems with deep learning

DeepInverse is an open-source PyTorch-based library for solving imaging inverse problems. The library covers all crucial steps in image reconstruction from the efficient implementation of forward operators (e.g., optics, MRI, tomography), to the definition and resolution of variational problems and the design and training of advanced neural network architectures. In this paper, we describe the main functionality of the library and discuss the main design choices.

eess.IV↗
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.

cs.LG↗
arXiv2024

Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure

Estimating matrices in the symmetric positive-definite (SPD) cone is of interest for many applications ranging from computer vision to graph learning. While there exist various convex optimization-based estimators, they remain limited in expressivity due to their model-based approach. The success of deep learning motivates the use of learning-based approaches to estimate SPD matrices with neural networks in a data-driven fashion. However, designing effective neural architectures for SPD learning is challenging, particularly when the task requires additional structural constraints, such as element-wise sparsity. Current approaches either do not ensure that the output meets all desired properties or lack expressivity. In this paper, we introduce SpodNet, a novel and generic learning module that guarantees SPD outputs and supports additional structural constraints. Notably, it solves the challenging task of learning jointly SPD and sparse matrices. Our experiments illustrate the versatility and relevance of SpodNet layers for such applications.

cs.LG↗
arXiv2023

Beyond L1: Faster and Better Sparse Models with skglm

We propose a new fast algorithm to estimate any sparse generalized linear model with convex or non-convex separable penalties. Our algorithm is able to solve problems with millions of samples and features in seconds, by relying on coordinate descent, working sets and Anderson acceleration. It handles previously unaddressed models, and is extensively shown to improve state-of-art algorithms. We provide a flexible, scikit-learn compatible package, which easily handles customized datafits and penalties.

stat.ML↗
arXiv2022

Benchopt: Reproducible, efficient and collaborative optimization benchmarks

Numerical validation is at the core of machine learning research as it allows to assess the actual impact of new methods, and to confirm the agreement between theory and practice. Yet, the rapid development of the field poses several challenges: researchers are confronted with a profusion of methods to compare, limited transparency and consensus on best practices, as well as tedious re-implementation work. As a result, validation is often very partial, which can lead to wrong conclusions that slow down the progress of research. We propose Benchopt, a collaborative framework to automate, reproduce and publish optimization benchmarks in machine learning across programming languages and hardware architectures. Benchopt simplifies benchmarking for the community by providing an off-the-shelf tool for running, sharing and extending experiments. To demonstrate its broad usability, we showcase benchmarks on three standard learning tasks: $\ell_2$-regularized logistic regression, Lasso, and ResNet18 training for image classification. These benchmarks highlight key practical findings that give a more nuanced view of the state-of-the-art for these problems, showing that for practical evaluation, the devil is in the details. We hope that Benchopt will foster collaborative work in the community hence improving the reproducibility of research findings.

cs.LG↗
arXiv2022

Coordinate Descent for SLOPE

The lasso is the most famous sparse regression and feature selection method. One reason for its popularity is the speed at which the underlying optimization problem can be solved. Sorted L-One Penalized Estimation (SLOPE) is a generalization of the lasso with appealing statistical properties. In spite of this, the method has not yet reached widespread interest. A major reason for this is that current software packages that fit SLOPE rely on algorithms that perform poorly in high dimensions. To tackle this issue, we propose a new fast algorithm to solve the SLOPE optimization problem, which combines proximal gradient descent and proximal coordinate descent steps. We provide new results on the directional derivative of the SLOPE penalty and its related SLOPE thresholding operator, as well as provide convergence guarantees for our proposed solver. In extensive benchmarks on simulated and real data, we show that our method outperforms a long list of competing algorithms.

math.OC↗
arXiv2022

Dual Extrapolation for Sparse Generalized Linear Models

Generalized Linear Models (GLM) form a wide class of regression and classification models, where prediction is a function of a linear combination of the input variables. For statistical inference in high dimension, sparsity inducing regularizations have proven to be useful while offering statistical guarantees. However, solving the resulting optimization problems can be challenging: even for popular iterative algorithms such as coordinate descent, one needs to loop over a large number of variables. To mitigate this, techniques known as screening rules and working sets diminish the size of the optimization problem at hand, either by progressively removing variables, or by solving a growing sequence of smaller problems. For both techniques, significant variables are identified thanks to convex duality arguments. In this paper, we show that the dual iterates of a GLM exhibit a Vector AutoRegressive (VAR) behavior after sign identification, when the primal problem is solved with proximal gradient descent or cyclic coordinate descent. Exploiting this regularity, one can construct dual points that offer tighter certificates of optimality, enhancing the performance of screening rules and helping to design competitive working set algorithms.

stat.ML↗
arXiv2022

Implicit differentiation for fast hyperparameter selection in non-smooth convex learning

Finding the optimal hyperparameters of a model can be cast as a bilevel optimization problem, typically solved using zero-order techniques. In this work we study first-order methods when the inner optimization problem is convex but non-smooth. We show that the forward-mode differentiation of proximal gradient descent and proximal coordinate descent yield sequences of Jacobians converging toward the exact Jacobian. Using implicit differentiation, we show it is possible to leverage the non-smoothness of the inner problem to speed up the computation. Finally, we provide a bound on the error made on the hypergradient when the inner optimization problem is solved approximately. Results on regression and classification problems reveal computational benefits for hyperparameter optimization, especially when multiple hyperparameters are required.

stat.ML↗
arXiv2022

Iterative regularization for low complexity regularizers

Iterative regularization exploits the implicit bias of an optimization algorithm to regularize ill-posed problems. Constructing algorithms with such built-in regularization mechanisms is a classic challenge in inverse problems but also in modern machine learning, where it provides both a new perspective on algorithms analysis, and significant speed-ups compared to explicit regularization. In this work, we propose and study the first iterative regularization procedure able to handle biases described by non smooth and non strongly convex functionals, prominent in low-complexity regularization. Our approach is based on a primal-dual algorithm of which we analyze convergence and stability properties, even in the case where the original problem is unfeasible. The general results are illustrated considering the special case of sparse recovery with the $\ell_1$ penalty. Our theoretical results are complemented by experiments showing the computational benefits of our approach.

math.OC↗
arXiv2021

Anderson acceleration of coordinate descent

Acceleration of first order methods is mainly obtained via inertial techniques à la Nesterov, or via nonlinear extrapolation. The latter has known a recent surge of interest, with successful applications to gradient and proximal gradient techniques. On multiple Machine Learning problems, coordinate descent achieves performance significantly superior to full-gradient methods. Speeding up coordinate descent in practice is not easy: inertially accelerated versions of coordinate descent are theoretically accelerated, but might not always lead to practical speed-ups. We propose an accelerated version of coordinate descent using extrapolation, showing considerable speed up in practice, compared to inertial accelerated coordinate descent and extrapolated (proximal) gradient descent. Experiments on least squares, Lasso, elastic net and logistic regression validate the approach.

stat.ML↗
arXiv2020

Iterative regularization for convex regularizers

We study iterative regularization for linear models, when the bias is convex but not necessarily strongly convex. We characterize the stability properties of a primal-dual gradient based approach, analyzing its convergence in the presence of worst case deterministic noise. As a main example, we specialize and illustrate the results for the problem of robust sparse recovery. Key to our analysis is a combination of ideas from regularization theory and optimization in the presence of errors. Theoretical results are complemented by experiments showing that state-of-the-art performances can be achieved with considerable computational speed-ups.

stat.ML↗
arXiv2020

Handling correlated and repeated measurements with the smoothed multivariate square-root Lasso

Sparsity promoting norms are frequently used in high dimensional regression. A limitation of such Lasso-type estimators is that the optimal regularization parameter depends on the unknown noise level. Estimators such as the concomitant Lasso address this dependence by jointly estimating the noise level and the regression coefficients. Additionally, in many applications, the data is obtained by averaging multiple measurements: this reduces the noise variance, but it dramatically reduces sample sizes and prevents refined noise modeling. In this work, we propose a concomitant estimator that can cope with complex noise structure by using non-averaged measurements. The resulting optimization problem is convex and amenable, thanks to smoothing theory, to state-of-the-art optimization techniques that leverage the sparsity of the solutions. Practical benefits are demonstrated on toy datasets, realistic simulated data and real neuroimaging data.

stat.ML↗