On the Accuracy of Hotelling-Type Tensor Deflation: A Random Tensor Analysis

TL;DR

This paper analyzes the accuracy of Hotelling-type tensor deflation in high-dimensional random tensor models, providing asymptotic characterizations of singular values and alignments to improve signal estimation.

Contribution

It offers the first asymptotic analysis of Hotelling-type tensor deflation for asymmetric spiked tensors, enabling consistent estimation of signal parameters.

Findings

Characterizes singular values and alignments in large dimensions

Provides methods for consistent estimation of signal-to-noise ratios

Analyzes the deflation procedure's accuracy asymptotically

Abstract

Leveraging on recent advances in random tensor theory, we consider in this paper a rank-$r$ asymmetric spiked tensor model of the form $\sum_{i=1}^r \beta_i A_i + W$ where $\beta_i\geq 0$ and the $A_i$'s are rank-one tensors such that $\langle A_i, A_j \rangle\in [0, 1]$ for $i\neq j$, based on which we provide an asymptotic study of Hotelling-type tensor deflation in the large dimensional regime. Specifically, our analysis characterizes the singular values and alignments at each step of the deflation procedure, for asymptotically large tensor dimensions. This can be used to construct consistent estimators of different quantities involved in the underlying problem, such as the signal-to-noise ratios $\beta_i$ or the alignments between the different signal components $\langle A_i, A_j \rangle$.