arXiv subjects

Irène Waldspurger

Publications and source records attributed to Irène Waldspurger.

18 recordsLinked to original sources

arXiv2026

Functional inequalities along Wasserstein geodesics

We study functional inequalities along Wasserstein geodesics. If $μ$ 0 and $μ$ 1 are respectively $κ$ 0 -and $κ$ 1 -strongly log-concave probability measures on R n , we prove that their quadratic Wasserstein geodesic satisfies for all probability measures $ν$ on R n . The coefficient is sharp. By linearization, this recovers the Poincar{é} estimate of Han and Zhu [15]. On the real line, we prove convexity of the square roots of the optimal T 1 and T 2 constants along monotone interpolation between arbitrary probability measures. The argument applies to more general transport entropy inequalities. We also establish convexity of the rescaled L p Poincar{é} constants for every finite p $\ge$ 1, including the square root of the Poincar{é} constant and the inverse Cheeger constant. Finally, we construct a planar Wasserstein geodesic whose endpoints satisfy T 2 and all finite-p L p -Poincar{é} inequalities, whereas every interior interpolant fails these inequalities.

math.FA↗

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↗

arXiv2026

Phase Transition in Convex Relaxations for Graph Alignment

We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+σ^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $σ= o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(Π^\star)$, i.e., $\|X^\star-Π^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-Π^\star\|_F^2$ transitions from $o(n)$ for $σ= \tilde{o}(n^{-1/2})$ to $Ω(n)$ for $σ= \tildeΩ(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.

stat.ML↗

arXiv2025

On the non-convexity issue in the radial Calderón problem

A classical approach to the Calderón problem is to estimate the unknown conductivity by solving a nonlinear least-squares problem. It leads to a nonconvex optimization problem which is generally believed to be riddled with bad local minimums. We revisit this issue in the case of piecewise constant radial conductivities and prove that, contrary to previous claims, there are no spurious critical points in the case of two scalar unknowns with no measurement noise. We also provide a partial proof of this result in the general setting which holds under a numerically verifiable assumption. Finally, we investigate whether a recently proposed approach based on convexification yields better reconstructions. For the first time, we propose a way to implement it in practice and show that it is consistently outperformed by some least squares solvers, which are also faster and require less measurements.

math.NA↗

arXiv2025

Graph Alignment via Birkhoff Relaxation

We consider the graph alignment problem, wherein the objective is to find a vertex correspondence between two graphs that maximizes the edge overlap. The graph alignment problem is an instance of the quadratic assignment problem (QAP), known to be NP-hard in the worst case even to approximately solve. In this paper, we analyze Birkhoff relaxation, a tight convex relaxation of QAP, and present theoretical guarantees on its performance when the inputs follow the Gaussian Wigner Model. More specifically, the weighted adjacency matrices are correlated Gaussian Orthogonal Ensemble with correlation $1/\sqrt{1+σ^2}$. Denote the optimal solutions of the QAP and Birkhoff relaxation by $Π^\star$ and $X^\star$ respectively. We show that $\|X^\star-Π^\star\|_F^2 = o(n)$ when $σ= o(n^{-1.25})$ and $\|X^\star-Π^\star\|_F^2 = Ω(n)$ when $σ= Ω(n^{-0.5})$. Thus, the optimal solution $X^\star$ transitions from a small perturbation of $Π^\star$ for small $σ$ to being well separated from $Π^\star$ as $σ$ becomes larger than $n^{-0.5}$. This result allows us to guarantee that simple rounding procedures on $X^\star$ align $1-o(1)$ fraction of vertices correctly whenever $σ= o(n^{-1.25})$. This condition on $σ$ to ensure the success of the Birkhoff relaxation is state-of-the-art.

stat.ML↗

arXiv2025

Projected gradient descent accumulates at Bouligand stationary points

This paper considers the projected gradient descent (PGD) algorithm for the problem of minimizing a continuously differentiable function on a nonempty closed subset of a Euclidean vector space. Without further assumptions, this problem is intractable and algorithms are only expected to find a stationary point. PGD generates a sequence in the set whose accumulation points are known to be Mordukhovich stationary. In this paper, these accumulation points are proven to be Bouligand stationary, and even proximally stationary if the gradient is locally Lipschitz continuous. These are the strongest stationarity properties that can be expected for the considered problem.

math.OC↗

arXiv2025

Benign landscape for Burer-Monteiro factorizations of MaxCut-type semidefinite programs

We consider MaxCut-type semidefinite programs (SDP) which admit a low rank solution. To numerically leverage the low rank hypothesis, a standard algorithmic approach is the Burer-Monteiro factorization, which allows to significantly reduce the dimensionality of the problem at the cost of its convexity. We give a sharp condition on the conditioning of the Laplacian matrix associated with the SDP under which any second-order critical point of the non-convex problem is a global minimizer. By applying our theorem, we improve on recent results about the correctness of the Burer-Monteiro approach on $\mathbb{Z}_2$-synchronization problems and the Kuramoto model.

math.OC↗

arXiv2021

Lecture notes on non-convex algorithms for low-rank matrix recovery

Low-rank matrix recovery problems are inverse problems which naturally arise in various fields like signal processing, imaging and machine learning. They are non-convex and NP-hard in full generality. It is therefore a delicate problem to design efficient recovery algorithms and to provide rigorous theoretical insights on the behavior of these algorithms. The goal of these notes is to review recent progress in this direction for the class of so-called "non-convex algorithms", with a particular focus on the proof techniques. Although they aim at presenting very recent research works, these notes have been written with the intent to be, as much as possible, accessible to non-specialists.

math.OC↗

arXiv2019

Rank optimality for the Burer-Monteiro factorization

When solving large scale semidefinite programs that admit a low-rank solution, an efficient heuristic is the Burer-Monteiro factorization: instead of optimizing over the full matrix, one optimizes over its low-rank factors. This reduces the number of variables to optimize, but destroys the convexity of the problem, thus possibly introducing spurious second-order critical points. The article [Boumal, Voroninski, and Bandeira, 2018] shows that when the size of the factors is of the order of the square root of the number of linear constraints, this does not happen: for almost any cost matrix, second-order critical points are global solutions. In this article, we show that this result is essentially tight: for smaller values of the size, second-order critical points are not generically optimal, even when the global solution is rank 1.

math.OC↗

arXiv2019

Phase retrieval with random Gaussian sensing vectors by alternating projections

We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a suitable initalization procedure, the classical algorithm of alternating projections succeeds with high probability when $m\geq Cn$, for some $C>0$. We conjecture that this result is still true when no special initialization procedure is used, and present numerical experiments that support this conjecture.

math.ST↗

arXiv2017

Phase retrieval for wavelet transforms

We describe a new algorithm to solve a particular phase retrieval problem, that has wide applications in audio processing: the reconstruction of a function from its scalogram, that is from the modulus of its wavelet transform. It is a multiscale iterative algorithm. To efficiently propagate phase information from low to high frequencies, it uses an equivalent formulation of the phase retrieval problem that involves the holomorphic extension of the wavelet transform. Our numerical results, on audio and non-audio signals, show that reconstruction is precise and stable to noise. The algorithm has a linear complexity in the size of the signal, up to logarithmic factors, and can thus be used on large signals.

math.OC↗

arXiv2017

Phase retrieval for the Cauchy wavelet transform

We consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform. We study the unicity and stability of the reconstruction. In the case where the wavelets are Cauchy wavelets, we prove that the modulus of the wavelet transform uniquely determines the function up to a global phase. We show that the reconstruction operator is continuous but not uniformly continuous. We describe how to construct pairs of functions which are far away in $L^2$-norm but whose wavelet transforms are very close, in modulus. The principle is to modulate the wavelet transform of a fixed initial function by a phase which varies slowly in both time and frequency. This construction seems to cover all the instabilities that we observe in practice; we give a partial formal justification to this fact. Finally, we describe an exact reconstruction algorithm and use it to numerically confirm our analysis of the stability question.

math.FA↗

arXiv2016

Phase retrieval with random Gaussian sensing vectors by alternating projections

We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors. We assume the sensing vectors to be independently sampled from complex normal distributions. We propose to solve this problem with the classical non-convex method of alternating projections. We show that, when $m\geq Cn$ for $C$ large enough, alternating projections succeed with high probability, provided that they are carefully initialized. We also show that there is a regime in which the stagnation points of the alternating projections method disappear, and the initialization procedure becomes useless. However, in this regime, $m$ has to be of the order of $n^2$. Finally, we conjecture from our numerical experiments that, in the regime $m=O(n)$, there are stagnation points, but the size of their attraction basin is small if $m/n$ is large enough, so alternating projections can succeed with probability close to $1$ even with no special initialization.

math.ST↗

arXiv2016

Exponential decay of scattering coefficients

We study an aspect of the following general question: which properties of a signal can be characterized by its scattering transform? We show that the energy contained in high order scattering coefficients is upper bounded by the energy contained in the high frequencies of the signal. This result links the decay of the scattering coefficients of a signal with the decay of its Fourier transform. Additionally, it allows to generalize some results of Mallat (2012), by relaxing the admissibility condition on the wavelet family.

math.FA↗

arXiv2015

Deep Learning by Scattering

We introduce general scattering transforms as mathematical models of deep neural networks with l2 pooling. Scattering networks iteratively apply complex valued unitary operators, and the pooling is performed by a complex modulus. An expected scattering defines a contractive representation of a high-dimensional probability distribution, which preserves its mean-square norm. We show that unsupervised learning can be casted as an optimization of the space contraction to preserve the volume occupied by unlabeled examples, at each layer of the network. Supervised learning and classification are performed with an averaged scattering, which provides scattering estimations for multiple classes.

cs.LG↗

arXiv2014

Phase retrieval for imaging problems

We study convex relaxation algorithms for phase retrieval on imaging problems. We show that structural assumptions on the signal and the observations, such as sparsity, smoothness or positivity, can be exploited to both speed-up convergence and improve recovery performance. We detail experimental results in molecular imaging problems simulated from PDB data.

math.OC↗

arXiv2013

Wavelet methods for shape perception in electro-sensing

This paper aims at presenting a new approach to the electro-sensing problem using wavelets. It provides an efficient algorithm for recognizing the shape of a target from micro-electrical impedance measurements. Stability and resolution capabilities of the proposed algorithm are quantified in numerical simulations.

math.NA↗

arXiv2013

Phase Recovery, MaxCut and Complex Semidefinite Programming

Phase retrieval seeks to recover a signal x from the amplitude |Ax| of linear measurements. We cast the phase retrieval problem as a non-convex quadratic program over a complex phase vector and formulate a tractable relaxation (called PhaseCut) similar to the classical MaxCut semidefinite program. We solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in Gerchberg-Saxton, where each iteration is a matrix vector product. Numerical results show the performance of this approach over three different phase retrieval problems, in comparison with greedy phase retrieval algorithms and matrix completion formulations.

math.OC↗