Permutation Recovery of Spikes in Noisy High-Dimensional Tensor Estimation

Gérard Ben Arous (), Cédric Gerbelot () and Vanessa Piccolo ()
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Gérard Ben Arous: New York University, Courant Institute of Mathematical Sciences
Cédric Gerbelot: New York University, Courant Institute of Mathematical Sciences
Vanessa Piccolo: Unité de Mathématiques Pures et Appliquées (UMPA), ENS Lyon

A chapter in Stochastic Analysis and Applications 2025, 2026, pp 425-476 from Springer

Abstract: Abstract We study the dynamics of gradient flow in high dimensions for the multi-spiked tensor problem, where the goal is to estimate $$r$$ r unknown signal vectors (spikes) from noisy Gaussian tensor observations. We analyze the maximum likelihood estimator, which corresponds to optimizing a high-dimensional, nonconvex random objective. Our main results determine the sample complexity and runtime required for gradient flow to efficiently recover all spikes, up to a permutation. We show that, during recovery, correlations between the estimators and true spikes increase sequentially, in an order depending on their initial value and on the associated signal-to-noise ratios (SNRs). This ordering determines the permutation under which the spikes are recovered. This work builds on our companion paper [4], which analyzes Langevin dynamics and establishes the sample complexity and SNR conditions required for exact recovery, where the recovered permutation matches the identity.

Keywords: High-dimensional optimization; Multi-spiked tensor PCA; Gradient flow dynamics; Permutation recovery; 68Q87; 62F30; 60H30 (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-03914-9_15

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DOI: 10.1007/978-3-032-03914-9_15

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