arXiv subjects

Charles Dossal

Publications and source records attributed to Charles Dossal.

At least 19 recordsLinked to original sources

arXiv2026

FlowID : Enhancing Forensic Identification with Latent Flow-Matching Models

Every day, many people die under violent circumstances, whether from crimes, war, migration, or climate disasters. Medico-legal and law enforcement institutions document many portraits of the deceased for evidence, but cannot immediately carry out identification on them. While traditional image editing tools can process these photos for public release, the workflow is lengthy and produces suboptimal results. In this work, we leverage advances in image generation models, which can now produce photorealistic human portraits, to introduce FlowID, an identity-preserving facial reconstruction method. Our approach combines single-image fine-tuning, which adapts the generative model to out-of-distribution injured faces, with attention-based masking that localizes edits to damaged regions while preserving identity-critical features. Together, these components enable the removal of artifacts from violent death while retaining sufficient identity information to support identification. To evaluate our method, we introduce InjuredFaces, a novel benchmark for identity-preserving facial reconstruction under severe facial damage. Beyond serving as an evaluation tool for this work, InjuredFaces provides a standardized resource for the community to study and compare methods addressing facial reconstruction in extreme conditions. Experimental results show that FlowID outperforms state-of-the-art open-source methods while maintaining low memory requirements, making it suitable for local deployment without compromising data privacy.

cs.CV↗

arXiv2026

Continuized Nesterov Momentum Achieves the $O(\varepsilon^{-7/4})$ Complexity in Smooth Nonconvex Optimization

For first-order optimization of non-convex functions with Lipschitz-continuous gradient and Hessian, the best-known complexity for reaching an $\varepsilon$-approximation of a stationary point is $\mathcal{O}(\varepsilon^{-7/4})$. The existing algorithms achieving this bound are based on momentum, but are always complemented with safeguard mechanisms that erase the accumulated momentum if a certain condition is violated. Whether such momentum-control mechanisms are fundamentally necessary has remained an open question. We show that randomizing the parameters enables one to achieve this complexity in expectation when using momentum without any of such mechanisms, and we improve the numerical constant factor of the bound in the case of a large enough number of iterations. From an analysis perspective, we do so by leveraging the continuized method, which interprets the algorithm as a realization of a continuous-time stochastic differential equation (SDE) involving a Poisson process. We show that this SDE converges in probability to the Heavy Ball ordinary differential equation when the stepsize goes to zero, paralleling the behavior of more classical instances of Nesterov momentum.

math.OC↗

arXiv2025

Continuized Nesterov Acceleration for Non-Convex Optimization

In convex optimization, continuous-time counterparts have been a fruitful tool for analyzing momentum algorithms. Fewer such examples are available when the function to minimize is non-convex. In several cases, discrepancies arise between the existing discrete-time results, namely those obtained for momentum algorithms, and their continuous-time counterparts, with the latter typically yielding stronger guarantees. We argue that the continuized framework (Even et al., 2021), mixing continuous and discrete components, can tighten the gap between known continuous and discrete results. This framework relies on computations akin to standard Lyapunov analyses, from which are deduced convergence bounds for an algorithm that can be written as a Nesterov momentum algorithm with stochastic parameters. In this work, we extend the range of applicability of the continuized framework, e.g. by allowing it to handle non-smooth Lyapunov functions. We then strengthen its trajectory-wise guarantees for linear convergence rate, deriving finite time bounds with high probability and asymptotic almost sure bounds. We apply this framework to the non-convex class of strongly quasar convex functions. Adapting continuous-time results that have weaker discrete equivalents to the continuized method, we improve by a constant factor the known convergence rate, and relax the existing assumptions on the set of minimizers.

math.OC↗

arXiv2024

Optimization with First Order Algorithms

These notes focus on the minimization of convex functionals using first-order optimization methods, which are fundamental in many areas of applied mathematics and engineering. The primary goal of this document is to introduce and analyze the most classical first-order optimization algorithms. We aim to provide readers with both a practical and theoretical understanding in how and why these algorithms converge to minimizers of convex functions. The main algorithms covered in these notes include gradient descent, Forward-Backward splitting, Douglas-Rachford splitting, the Alternating Direction Method of Multipliers (ADMM), and Primal-Dual algorithms. All these algorithms fall into the class of first order methods, as they only involve gradients and subdifferentials, that are first order derivatives of the functions to optimize. For each method, we provide convergence theorems, with precise assumptions and conditions under which the convergence holds, accompanied by complete proofs. Beyond convex optimization, the final part of this manuscript extends the analysis to nonconvex problems, where we discuss the convergence behavior of these same first-order methods under broader assumptions. To contextualize the theory, we also include a selection of practical examples illustrating how these algorithms are applied in different image processing problems.

math.OC↗

arXiv2024

Gradient correlation is a key ingredient to accelerate SGD with momentum

Empirically, it has been observed that adding momentum to Stochastic Gradient Descent (SGD) accelerates the convergence of the algorithm. However, the literature has been rather pessimistic, even in the case of convex functions, about the possibility of theoretically proving this observation. We investigate the possibility of obtaining accelerated convergence of the Stochastic Nesterov Accelerated Gradient (SNAG), a momentum-based version of SGD, when minimizing a sum of functions in a convex setting. We demonstrate that the average correlation between gradients allows to verify the strong growth condition, which is the key ingredient to obtain acceleration with SNAG. Numerical experiments, both in linear regression and deep neural network optimization, confirm in practice our theoretical results.

math.OC↗

arXiv2024

Strong Convergence of FISTA Iterates under H{\"o}lderian and Quadratic Growth Conditions

Introduced by Beck and Teboulle, FISTA (for Fast Iterative Shrinkage-Thresholding Algorithm) is a first-order method widely used in convex optimization. Adapted from Nesterov's accelerated gradient method for convex functions, the generated sequence guarantees a decay of the function values of $\mathcal{O}\left(n^{-2}\right)$ in the convex setting. We show that for coercive functions satisfying some local growth condition (namely a H\''olderian or quadratic growth condition), this sequence strongly converges to a minimizer. This property, which has never been proved without assuming the uniqueness of the minimizer, is associated with improved convergence rates for the function values. The proposed analysis is based on a preliminary study of the Asymptotic Vanishing Damping system introduced by Su et al. in to modelNesterov's accelerated gradient method in a continuous setting. Novel improved convergence results are also shown for the solutions of this dynamical system, including the finite length of the trajectory under the aforementioned geometry conditions.

math.OC↗

arXiv2024

Study of the behaviour of Nesterov Accelerated Gradient in a non convex setting: the strongly quasar convex case

We study the convergence of Nesterov Accelerated Gradient (NAG) minimization algorithmapplied to a class of non convex functions called strongly quasar convex functions. We show thatNAG can achieve an accelerated convergence speed at the cost of a lower curvature assumption.We provide a continuous analysis through high resolution ODEs, where we show that despite thatnegative friction may appear, the solution of the system achieves accelerated rate of convergenceto the minimum. Finally, we identify the key geometrical property that, if dropped, theoreticallycancels the acceleration phenomenon.

math.OC↗

arXiv2024

Heavy Ball Momentum for Non-Strongly Convex Optimization

When considering the minimization of a quadratic or strongly convex function, it is well known that first-order methods involving an inertial term weighted by a constant-in-time parameter are particularly efficient (see Polyak [32], Nesterov [28], and references therein). By setting the inertial parameter according to the condition number of the objective function, these methods guarantee a fast exponential decay of the error. We prove that this type of schemes (which are later called Heavy Ball schemes) is relevant in a relaxed setting, i.e. for composite functions satisfying a quadratic growth condition. In particular, we adapt V-FISTA, introduced by Beck in [10] for strongly convex functions, to this broader class of functions. To the authors' knowledge, the resulting worst-case convergence rates are faster than any other in the literature, including those of FISTA restart schemes. No assumption on the set of minimizers is required and guarantees are also given in the non-optimal case, i.e. when the condition number is not exactly known. This analysis follows the study of the corresponding continuous-time dynamical system (Heavy Ball with friction system), for which new convergence results of the trajectory are shown.

math.OC↗

arXiv2023

Parameter-Free FISTA by Adaptive Restart and Backtracking

We consider a combined restarting and adaptive backtracking strategy for the popular Fast Iterative Shrinking-Thresholding Algorithm frequently employed for accelerating the convergence speed of large-scale structured convex optimization problems. Several variants of FISTA enjoy a provable linear convergence rate for the function values $F(x_n)$ of the form $\mathcal{O}( e^{-K\sqrt{\mu/L}~n})$ under the prior knowledge of problem conditioning, i.e. of the ratio between the (\L ojasiewicz) parameter $\mu$ determining the growth of the objective function and the Lipschitz constant $L$ of its smooth component. These parameters are nonetheless hard to estimate in many practical cases. Recent works address the problem by estimating either parameter via suitable adaptive strategies. In our work both parameters can be estimated at the same time by means of an algorithmic restarting scheme where, at each restart, a non-monotone estimation of $L$ is performed. For this scheme, theoretical convergence results are proved, showing that a $\mathcal{O}( e^{-K\sqrt{\mu/L}n})$ convergence speed can still be achieved along with quantitative estimates of the conditioning. The resulting Free-FISTA algorithm is therefore parameter-free. Several numerical results are reported to confirm the practical interest of its use in many exemplar problems.

math.OC↗

arXiv2022

Fast convergence of inertial dynamics with Hessian-driven damping under geometry assumptions

First-order optimization algorithms can be considered as a discretization of ordinary differential equations (ODEs) \cite{su2014differential}. In this perspective, studying the properties of the corresponding trajectories may lead to convergence results which can be transfered to the numerical scheme. In this paper we analyse the following ODE introduced by Attouch et al. in \cite{attouch2016fast}: \begin{equation*} \forall t\geqslant t_0,~\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\beta H_F(x(t))\dot{x}(t)+\nabla F(x(t))=0,\end{equation*} where $\alpha>0$, $\beta>0$ and $H_F$ denotes the Hessian of $F$. This ODE can be derived to build numerical schemes which do not require $F$ to be twice differentiable as shown in \cite{attouch2020first,attouch2021convergence}. We provide strong convergence results on the error $F(x(t))-F^*$ and integrability properties on $\|\nabla F(x(t))\|$ under some geometry assumptions on $F$ such as quadratic growth around the set of minimizers. In particular, we show that the decay rate of the error for a strongly convex function is $O(t^{-\alpha-\varepsilon})$ for any $\varepsilon>0$. These results are briefly illustrated at the end of the paper.

math.OC↗

arXiv2019

Nesterov's acceleration and Polyak's heavy ball method in continuous time: convergence rate analysis under geometric conditions and perturbations

In this article a family of second order ODEs associated to inertial gradient descend is studied. These ODEs are widely used to build trajectories converging to a minimizer $x^*$ of a function $F$, possibly convex. This family includes the continuous version of the Nesterov inertial scheme and the continuous heavy ball method. Several damping parameters, not necessarily vanishing, and a perturbation term $g$ are thus considered. The damping parameter is linked to the inertia of the associated inertial scheme and the perturbation term $g$ is linked to the error that can be done on the gradient of the function $F$. This article presents new asymptotic bounds on $F(x(t))-F(x^*)$ where $x$ is a solution of the ODE, when $F$ is convex and satisfies local geometrical properties such as {\L}ojasiewicz properties and under integrability conditions on $g$. Even if geometrical properties and perturbations were already studied for most ODEs of these families, it is the first time they are jointly studied. All these results give an insight on the behavior of these inertial and perturbed algorithms if $F$ satisfies some {\L}ojasiewicz properties especially in the setting of stochastic algorithms.

math.OC↗

arXiv2018

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 \L ojasiewicz 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↗

arXiv2017

Overrelaxed Sinkhorn-Knopp Algorithm for Regularized Optimal Transport

This article describes a set of methods for quickly computing the solution to the regularized optimal transport problem. It generalizes and improves upon the widely-used iterative Bregman projections algorithm (or Sinkhorn--Knopp algorithm). We first propose to rely on regularized nonlinear acceleration schemes. In practice, such approaches lead to fast algorithms, but their global convergence is not ensured. Hence, we next propose a new algorithm with convergence guarantees. The idea is to overrelax the Bregman projection operators, allowing for faster convergence. We propose a simple method for establishing global convergence by ensuring the decrease of a Lyapunov function at each step. An adaptive choice of overrelaxation parameter based on the Lyapunov function is constructed. We also suggest a heuristic to choose a suitable asymptotic overrelaxation parameter, based on a local convergence analysis. Our numerical experiments show a gain in convergence speed by an order of magnitude in certain regimes.

math.NA↗

arXiv2016

Sampling the Fourier transform along radial lines

This article considers the use of total variation minimization for the recovery of a superposition of point sources from samples of its Fourier transform along radial lines. We present a numerical algorithm for the computation of solutions to this infinite dimensional problem. The theoretical results of this paper make precise the link between the sampling operator and the recoverability of the point sources.

math.NA↗

arXiv2014

The Degrees of Freedom of Partly Smooth Regularizers

In this paper, we are concerned with regularized regression problems where the prior regularizer is a proper lower semicontinuous and convex function which is also partly smooth relative to a Riemannian submanifold. This encompasses as special cases several known penalties such as the Lasso ($\ell^1$-norm), the group Lasso ($\ell^1-\ell^2$-norm), the $\ell^\infty$-norm, and the nuclear norm. This also includes so-called analysis-type priors, i.e. composition of the previously mentioned penalties with linear operators, typical examples being the total variation or fused Lasso penalties.We study the sensitivity of any regularized minimizer to perturbations of the observations and provide its precise local parameterization.Our main sensitivity analysis result shows that the predictor moves locally stably along the same active submanifold as the observations undergo small perturbations. This local stability is a consequence of the smoothness of the regularizer when restricted to the active submanifold, which in turn plays a pivotal role to get a closed form expression for the variations of the predictor w.r.t. observations. We also show that, for a variety of regularizers, including polyhedral ones or the group Lasso and its analysis counterpart, this divergence formula holds Lebesgue almost everywhere.When the perturbation is random (with an appropriate continuous distribution), this allows us to derive an unbiased estimator of the degrees of freedom and of the risk of the estimator prediction.Our results hold true without requiring the design matrix to be full column rank.They generalize those already known in the literature such as the Lasso problem, the general Lasso problem (analysis $\ell^1$-penalty), or the group Lasso where existing results for the latter assume that the design is full column rank.

math.ST↗

arXiv2012

The degrees of freedom of the Group Lasso for a General Design

In this paper, we are concerned with regression problems where covariates can be grouped in nonoverlapping blocks, and where only a few of them are assumed to be active. In such a situation, the group Lasso is an at- tractive method for variable selection since it promotes sparsity of the groups. We study the sensitivity of any group Lasso solution to the observations and provide its precise local parameterization. When the noise is Gaussian, this allows us to derive an unbiased estimator of the degrees of freedom of the group Lasso. This result holds true for any fixed design, no matter whether it is under- or overdetermined. With these results at hand, various model selec- tion criteria, such as the Stein Unbiased Risk Estimator (SURE), are readily available which can provide an objectively guided choice of the optimal group Lasso fit.

cs.IT↗

arXiv2012

Consistency of l1 recovery from noisy deterministic measurements

In this paper a new result of recovery of sparse vectors from deterministic and noisy measurements by l1 minimization is given. The sparse vector is randomly chosen and follows a generic p-sparse model introduced by Candes and al. The main theorem ensures consistency of l1 minimization with high probability. This first result is secondly extended to compressible vectors.

math.OC↗

arXiv2012

Risk estimation for matrix recovery with spectral regularization

In this paper, we develop an approach to recursively estimate the quadratic risk for matrix recovery problems regularized with spectral functions. Toward this end, in the spirit of the SURE theory, a key step is to compute the (weak) derivative and divergence of a solution with respect to the observations. As such a solution is not available in closed form, but rather through a proximal splitting algorithm, we propose to recursively compute the divergence from the sequence of iterates. A second challenge that we unlocked is the computation of the (weak) derivative of the proximity operator of a spectral function. To show the potential applicability of our approach, we exemplify it on a matrix completion problem to objectively and automatically select the regularization parameter.

math.OC↗