arXiv subjects

Thibault Pautrel

Publications and source records attributed to Thibault Pautrel.

5 recordsLinked to original sources

arXiv2026

Global universality of the expected number of zeros of non-analytic random signals

We study the asymptotics as $n$ goes to infinity of $\mathbb E\left[\mathcal{N}(S_n,[0,2\pi])\right]$, the expected number of zeros in $[0, 2\pi]$ of a random periodic signal $S_n$ of the form \[ S_n(t)=\sum_{k=1}^{n}a_k f(kt), \] where $f$ is a non-analytic $2\pi-$periodic function and the coefficients $(a_k)$ are i.i.d. random variables, centered with unit variance. We show in particular that if $a_1$ admits a finite third moment and if the function $f$ is piecewise polynomials and of class $\mathcal C^{7}$, then we have the following universal asymptotics, independent of the particular law of the coefficients $(a_k)$ \[ \lim_{n \to +\infty}\frac{\Esp\left[\mathcal{N}(S_n,[0,2\pi])\right]}{n}= \frac{2}{\sqrt{3}}\sqrt{\frac{\|f'\|_{L^2([0,2\pi])}}{\|f\|_{L^2([0,2\pi])}}}. \] This result thus extends in expectation and at the scale of the whole period $[0,2\pi]$ the local universality property established {in [Angst-Poly, IMRN, 2019]}, in distribution and in shrinking intervals of size $1/n$. Moreover, it generalizes to a non-analytic context the global universality results obtained in the more classical frameworks of random trigonometric polynomials or random analytic functions. Our approach combines a new almost sure Central Limit Theorem \`a la Salem--Zygmund for the function $S_n$ when evaluated at a uniform random point in $[0, 2\pi]$, and as well as suitable uniform integrability and anti-concentration estimates.

math.PR↗

arXiv2026

Riemannian Stochastic Optimization for Sufficient Dimension Reduction

Sufficient dimension reduction (SDR) makes high-dimensional regression tractable by projecting the covariates onto a low-dimensional subspace that preserves the conditional mean of the response. Existing gradient-based estimators either operate in the ambient space and suffer from the curse of dimensionality, or localize in the reduced space at a per-outer-iteration cost at least quadratic in the sample size. We show that minimizers of the population Minimum Average Variance Estimation (MAVE) risk approximate the same Grassmannian target as the Outer Product of Gradients (OPG), and recast the empirical criterion as a smooth maximization on the Stiefel manifold with closed-form Riemannian gradient. The resulting algorithm, SMAVE, combines sparse projected-space nearest-neighbor localization with Riemannian stochastic gradient ascent. A simplified version comes with almost-sure convergence and a non-asymptotic rate matching the standard non-convex stochastic first-order scaling. Empirically, SMAVE matches or improves on RMAVE's synthetic subspace recovery at moderate-to-high ambient dimension, and on four real datasets it uniformly improves over OPG and is competitive with or outperforms RMAVE at orders of magnitude lower runtime.

stat.ML↗

arXiv2026

FedSPDnet: Geometry-Aware Federated Deep Learning with SPDnet

We introduce two federated learning frameworks for the classical SPDnet model operating on symmetric positive definite (SPD) matrices with Stiefel-constrained parameters. Unlike standard Euclidean averaging, which violates orthogonality, our approach preserves geometric structure through two efficient aggregation strategies: ProjAvg, projecting arithmetic means onto the Stiefel manifold, and RLAvg, approximating tangent-space averaging via retractions and liftings. Both methods are computationally efficient, independent of the optimizer, and enable scalable federated learning for signal processing applications whose features are SPD matrices. Simulations on EEG motor imagery benchmarks show that FedSPDnet outperforms federated EEGnet in F1 score and robustness to federation and partial participation, while using fewer parameters per communication round.

stat.ML↗

arXiv2021

Real zeros of random trigonometric polynomials with dependent coefficients

We further investigate the relations between the large degree asymptotics of the number of real zeros of random trigonometric polynomials with dependent coefficients and the underlying correlation function. We consider trigonometric polynomials of the form \[ f_n(t):= \frac{1}{\sqrt{n}}\sum_{k=1}^{n}a_k \cos(kt)+b_k\sin(kt), ~x\in [0,2π], \] where the sequences $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent copies of a stationary Gaussian process centered with variance one and correlation function $ρ$ with associated spectral measure $μ_ρ$. We focus here on the case where $μ_ρ$ is not purely singular and we denote by $ψ_ρ$ its density component with respect to the Lebesgue measure $λ$. Quite surprisingly, we show that the asymptotics of the number of real zeros $\mathcal{N}(f_n,[0,2π])$ of $f_n$ in $[0,2π]$ is not related to the decay of the correlation function $ρ$ but instead to the Lebesgue measure of the vanishing locus of $ψ_ρ$. Namely, assuming that $ψ_ρ$ is $\mathcal{C}^1$ with Hölder derivative on an open set of full measure, one establishes that \[ \lim_{n \to +\infty} \frac{\mathbb E\left[\mathcal{N}(f_n,[0,2π])\right]}{n}= \frac{λ(\{ψ_ρ=0\})}{π\sqrt{2}} + \frac{2π- λ(\{ψ_ρ=0\})}{π\sqrt{3}}. \] On the other hand, assuming a sole log-integrability condition on $ψ_ρ$, which implies that it is positive almost everywhere, we recover the asymptotics of the independent case, i.e. the limit is $\frac{2}{\sqrt{3}}$. Besides, with further assumptions of regularity and existence of negative moment for $ψ_ρ$, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence.

math.PR↗

arXiv2020

New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients

We consider random trigonometric polynomials of the form \[ f_n(t):=\frac{1}{\sqrt{n}} \sum_{k=1}^{n}a_k \cos(k t)+b_k \sin(k t), \] where $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent stationary Gaussian processes with the same correlation function $ρ: k \mapsto \cos(kα)$, with $α\geq 0$. We show that the asymptotics of the expected number of real zeros differ from the universal one $\frac{2}{\sqrt{3}}$, holding in the case of independent or weakly dependent coefficients. More precisely, for all $\varepsilon>0$, for all $\ell \in (\sqrt{2},2]$, there exists $α\geq 0$ and $n\geq 1$ large enough such that $$\left|\frac{\mathbb{E}\left[\mathcal{N}(f_n,[0,2π])\right]}{n}-\ell\right|\leq \varepsilon,$$ where $\mathcal N(f_n,[0,2π])$ denotes the number of real zeros of the function $f_n$ in the interval $[0,2π]$. Therefore, this result provides the first example where the expected number of real zeros do not converge as $n$ goes to infinity by exhibiting a whole range of possible limits ranging from $\sqrt{2}$ to 2.

math.PR↗