arXiv subjects

Jean François Aujol

Publications and source records attributed to Jean François Aujol.

4 recordsLinked to original sources

arXiv2026

Towards optimal algorithms for the recovery of low-dimensional models with linear rates

We consider the problem of recovering elements of a low-dimensional model from linear measurements. From signal and image processing to inverse problems in data science, this question has been at the center of many applications. Lately, with the success of models and methods relying on deep neural networks, there has been a multiplication of different algorithms and recovery results. Comparing the performance of recovery algorithms becomes a complex task without a unifying framework. In this article, as a first step for the study of general algorithms for low-dimensional recovery, we study a class of generalized projected gradient descent algorithms that can recover a given low-dimensional model with linear rates. The obtained rates decouple the impact of the quality of the measurements with respect to the model from the geometry of the properties of the chosen generalized projection: we can directly measure performance through a restricted Lipschitz constant of the projection with respect to the low dimensional model. By optimizing this constant, we define an optimal generalized projected gradient descent. Our general approach provides an optimality result in the case of sparse recovery. Moreover, our framework allows for a common interpretation of linear rates of recovery in the context of both sparse models and models induced by some ``plug-and-play'' imaging methods that rely on deep neural networks. These rates of recovery are observed in experiments on synthetic and real data.

eess.SP↗

arXiv2025

On the impact of the parametrization of deep convolutional neural networks on post-training quantization

This paper introduces novel theoretical approximation bounds for the output of quantized neural networks, with a focus on convolutional neural networks (CNN). By considering layerwise parametrization and focusing on the quantization of weights, we provide bounds that gain several orders of magnitude compared to state-of-the-art results on classical deep convolutional neural networks such as MobileNetV2 or ResNets. These gains are achieved by improving the behaviour of the approximation bounds with respect to the depth parameter, which has the most impact on the approximation error induced by quantization. To complement our theoretical result, we provide a numerical exploration of our bounds on MobileNetV2 and ResNets.

cs.IT↗

arXiv2024

Projected Block Coordinate Descent for sparse spike estimation

We consider the problem of recovering off-the-grid spikes from linear measurements. The state of the art Over-Parametrized Continuous Orthogonal Matching Pursuit (OP-COMP) with Projected Gradient Descent (PGD) successfully recovers those signals. In most cases, the main computational cost lies in a unique global descent on all parameters (positions and amplitudes). In this paper, we propose to improve this algorithm by accelerating this descent step. We introduce a new algorithm, based on Block Coordinate Descent, that takes advantages of the sparse structure of the problem. Based on qualitative theoretical results, this algorithm shows improvement in calculation times in realistic synthetic microscopy experiments.

math.NA↗

arXiv2019

Optimal convergence rates for Nesterov acceleration

In this paper, we study the behavior of solutions of the ODE associated to Nesterov acceleration. It is well-known since the pioneering work of Nesterov that the rate of convergence $O(1/t^2)$ is optimal for the class of convex functions with Lipschitz gradient. In this work, we show that better convergence rates can be obtained with some additional geometrical conditions, such as Łojasiewicz property. More precisely, we prove the optimal convergence rates that can be obtained depending on the geometry of the function $F$ to minimize. The convergence rates are new, and they shed new light on the behavior of Nesterov acceleration schemes. We prove in particular that the classical Nesterov scheme may provide convergence rates that are worse than the classical gradient descent scheme on sharp functions: for instance, the convergence rate for strongly convex functions is not geometric for the classical Nesterov scheme (while it is the case for the gradient descent algorithm). This shows that applying the classical Nesterov acceleration on convex functions without looking more at the geometrical properties of the objective functions may lead to sub-optimal algorithms.

math.OC↗