Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2507.08444

Effective regions and kernels in continuous sparse regularisation, with application to sketched mixtures

Yohann De Castro·Rémi Gribonval·Nicolas Jouvin
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper advances the general theory of continuous sparse regularisation on measures with the Beurling-LASSO (BLASSO). This TV-regularised convex program on the space of measures allows to recover a sparse measure using a noisy observation from a measurement operator. While previous works have uncovered the central role played by this operator and its associated kernel in order to get estimation error bounds, the latter requires a technical local positive curvature (LPC) assumption to be verified on a case-by-case basis. In practice, this yields only few LPC-kernels for which this condition is proved. In this paper, we prove that the ``sinc-4'' kernel, used for signal recovery and mixture problems, does satisfy the LPC assumption. Furthermore, we introduce the kernel switch analysis, which allows to leverage on a known LPC-kernel as a pivot kernel to prove error bounds. Together, these results provide easy-to-check conditions to get error bounds for a large family of translation-invariant model kernels. Besides, we also show that known BLASSO guarantees can be made adaptive to the noise level. This improves on known results where this error is fixed with some parameters depending on the model kernel. We illustrate the interest of our results in the case of mixture model estimation, using band-limiting smoothing and sketching techniques to reduce the computational burden of BLASSO.

Explore related subjects

math.STstat.MLstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yohann De Castro, Rémi Gribonval, Nicolas Jouvin. 2025-10-14. Effective regions and kernels in continuous sparse regularisation, with application to sketched mixtures. https://arxiv.org/abs/2507.08444

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sample complexity and weak limits of nonsmooth multimarginal Schrödinger system with application to optimal transport barycenter

Multimarginal optimal transport (MOT) has emerged as a useful framework for many applied problems. However, compared to the well-studied classical two-marginal optimal transport theory, analysis of MOT is far more challenging and remains much less developed. In this paper, we study the statistical estimation and inference problems for the entropic MOT (EMOT), whose optimal solution is characterized by the multimarginal Schrödinger system. Assuming only boundedness of the cost function, we derive sharp sample complexity for estimating several key quantities pertaining to EMOT (cost functional and Schrödinger coupling) from point clouds that are randomly sampled from the input marginal distributions. Moreover, with substantially weaker smoothness assumption on the cost function than the existing literature, we derive distributional limits and bootstrap validity of various key EMOT objects. As an application, we propose the multimarginal Schrödinger barycenter as a new and natural way to regularize the exact Wasserstein barycenter and demonstrate its statistical optimality.

math.ST↗
arXiv2026

Nonparametric spectral density estimation using interactive mechanisms under local differential privacy

We study the problem of estimating the spectral density of a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: recovering a single covariance coefficient, recovering the spectral density at a fixed frequency, and global recovery. Our approach achieves faster rates through a two-stage process: we first apply the Laplace mechanism to the truncated value, and then use the resulting privatized sample to learn about the dependence mechanism in the time series. For spectral densities belonging to Hölder and Sobolev smoothness classes, we demonstrate that our algorithms improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter $α$, since the pointwise rates depend on $nα^2$ instead of $nα^4$. Moreover, we show that the rate $(nα^4)^{-1}$ is optimal for estimating a covariance coefficient with non-interactive mechanisms. However, the $L_2$ rate of our interactive estimator is slower than the pointwise rate. We show how to use these procedures to provide a bona fide locally differentially private estimator of the entire covariance matrix. A simulation study validates our findings.

math.ST↗
arXiv2026

Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency

Let $X = [ ξ_1, \,\, ξ_2,...\,\, ,ξ_d]^\top$ be a zero-mean random vector of large dimension $d$ ($d \rightarrow \infty$) with (hidden) covariance matrix $M = (m_{ij})_{1 \leq i, j \leq d},$ where $m_{ij} = m_{ji} = \textbf{Cov}(ξ_i, ξ_j).$ Let $X_1, X_2, \dots, X_n$ be $n$ iid samples of $X$. Consider the sample covariance matrix $$\textstyle \tilde{M} := \frac{1}{n} \sum_{i=1}^{n} X_i X_i^\top.$$ In practice, one frequently uses the eigenvectors and eigenspaces of $\tilde M$ as estimators for those of $M$. A central task is to provide an error analysis for these estimators. In this paper, we provide an optimal error analysis, obtaining upper and lower bounds of matching order of magnitude, for a wide range of parameters $d$ and $n$, under mild assumptions on $M$. As corollaries, we obtain new necessary and sufficient conditions for the consistency of the estimators. In these conditions, we only require the number of samples $n$ to depend linearly on the effective rank of $M$, which can be much smaller than the dimension $d$.

math.ST↗