Statistical Inference for Low-Rank Tensor Models

TL;DR

This paper develops a unified statistical inference framework for low-rank tensor models, enabling asymptotic normality and optimal confidence intervals for general linear functionals in high-dimensional tensor data.

Contribution

It introduces a debiasing and projection method for inference in low-Tucker-rank tensors, extending beyond entrywise analysis and achieving near-optimal sample and SNR thresholds.

Findings

Achieves asymptotic normality for linear functionals of low-rank tensors.

Constructs minimax-optimal confidence intervals without sparsity assumptions.

Validates the framework through numerical experiments in various applications.

Abstract

Statistical inference for tensors has emerged as a critical challenge in analyzing high-dimensional data in modern data science. This paper introduces a unified framework for inferring general and low-Tucker-rank linear functionals of low-Tucker-rank signal tensors for several low-rank tensor models. Our methodology tackles two primary goals: achieving asymptotic normality and constructing minimax-optimal confidence intervals. By leveraging a debiasing strategy and projecting onto the tangent space of the low-Tucker-rank manifold, we enable inference for general and structured linear functionals, extending far beyond the scope of traditional entrywise inference. Specifically, in the low-Tucker-rank tensor regression or PCA model, we establish the computational and statistical efficiency of our approach, achieving near-optimal sample size requirements (in regression model) and signal-to-noise ratio (SNR) conditions (in PCA model) for general linear functionals without requiring sparsity in the loading tensor. Our framework also attains both computationally and statistically optimal sample size and SNR thresholds for low-Tucker-rank linear functionals. Numerical experiments validate our theoretical results, showcasing the framework's utility in diverse applications. This work addresses significant methodological gaps in statistical inference, advancing tensor analysis for complex and high-dimensional data environments.