Inference for Low-rank Tensors -- No Need to Debias

TL;DR

This paper develops methods for statistical inference on low-rank tensors, establishing asymptotic distributions and confidence regions without the need for debiasing, under certain conditions related to signal strength or sample size.

Contribution

It introduces a novel inference framework for low-rank tensors that avoids debiasing, leveraging conditions that ensure feasible estimation and asymptotic normality.

Findings

Confidence regions for tensor singular subspaces are constructed.

Asymptotic distributions are derived under specific signal-to-noise or sample size conditions.

Debiasing is unnecessary for inference in low-rank tensor models under these conditions.

Abstract

In this paper, we consider the statistical inference for several low-rank tensor models. Specifically, in the Tucker low-rank tensor PCA or regression model, provided with any estimates achieving some attainable error rate, we develop the data-driven confidence regions for the singular subspace of the parameter tensor based on the asymptotic distribution of an updated estimate by two-iteration alternating minimization. The asymptotic distributions are established under some essential conditions on the signal-to-noise ratio (in PCA model) or sample size (in regression model). If the parameter tensor is further orthogonally decomposable, we develop the methods and non-asymptotic theory for inference on each individual singular vector. For the rank-one tensor PCA model, we establish the asymptotic distribution for general linear forms of principal components and confidence interval for each entry of the parameter tensor. Finally, numerical simulations are presented to corroborate our theoretical discoveries. In all these models, we observe that different from many matrix/vector settings in existing work, debiasing is not required to establish the asymptotic distribution of estimates or to make statistical inference on low-rank tensors. In fact, due to the widely observed statistical-computational-gap for low-rank tensor estimation, one usually requires stronger conditions than the statistical (or information-theoretic) limit to ensure the computationally feasible estimation is achievable. Surprisingly, such conditions ``incidentally" render a feasible low-rank tensor inference without debiasing.