Permutation recovery of spikes in noisy high-dimensional tensor estimation

TL;DR

This paper analyzes the gradient flow dynamics in high-dimensional tensor estimation to determine the sample complexity needed for accurately recovering multiple unknown signals from noisy data, without requiring SNR separation assumptions.

Contribution

It provides the first analysis of the sample complexity for gradient flow in multi-spiked tensor problems, extending previous work on Langevin dynamics and exact recovery conditions.

Findings

Gradient flow can efficiently recover all spikes under certain sample complexity conditions.

Recovery order depends on initial correlations and SNRs, affecting permutation matching.

No assumptions on SNR separation are needed for successful recovery.

Abstract

We study the dynamics of gradient flow in high dimensions for the multi-spiked tensor problem, where the goal is to estimate $r$ unknown signal vectors (spikes) from noisy Gaussian tensor observations. Specifically, we analyze the maximum likelihood estimation procedure, which involves optimizing a highly nonconvex random function. We determine the sample complexity required for gradient flow to efficiently recover all spikes, without imposing any assumptions on the separation of the signal-to-noise ratios (SNRs). More precisely, our results provide the sample complexity required to guarantee recovery of the spikes up to a permutation. Our work builds on our companion paper [Ben Arous, Gerbelot, Piccolo 2024], which studies Langevin dynamics and determines the sample complexity and separation conditions for the SNRs necessary for ensuring exact recovery of the spikes (where the recovered permutation matches the identity). During the recovery process, the correlations between the estimators and the hidden vectors increase in a sequential manner. The order in which these correlations become significant depends on their initial values and the corresponding SNRs, which ultimately determines the permutation of the recovered spikes.