Statistical Inference for Low-Rank Tensor Models

TL;DR

The paper develops a unified debiased-tangent-space approach to statistical inference for general and low-Tucker-rank linear functionals of tensors in tensor regression and tensor PCA. By projecting debiased estimators onto the tangent space of the low-Tucker-rank manifold and (when desirable) employing sample splitting, it achieves asymptotic normality and minimax-optimal confidence-interval lengths under weakened incoherence conditions and practical sub-Gaussian assumptions. The work provides explicit sample-size and SNR thresholds for both general and low-rank linear functionals, deriving data-driven procedures for variance estimation and CI construction. Numerical experiments confirm the theoretical guarantees and demonstrate applicability across diverse tensor models and loading structures, offering a pathway to scalable, uncertainty-aware tensor inference in high-dimensional data settings.

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.

Figures (2)

• Figure 1: Histogram of normal approximation under the tensor regression setting based on 1000 independent replications, with $\overline{p}=40$ and $\overline{r}=3$. For single-entry inference, $n \in \{2\overline{p}^{3/4}\overline{r}, 2\overline{p}\overline{r}, 2\overline{p}^{5/4}\overline{r}\}$. For low-Tucker-rank linear functional inference, $n \in \{2\overline{p}^{5/4}\overline{r}, 2\overline{p}^{3/2}\overline{r}, 2\overline{p}^{7/4}\overline{r}\}$. For general linear functional inference, $n \in \{\overline{p}^{7/4}\overline{r}, \overline{p}^{2}\overline{r}, \overline{p}^{9/4}\overline{r}\}$.
• Figure 2: Histogram of normal approximation under the tensor PCA setting based on 1000 independent replications, with $\overline{p}=100$ and $\overline{r}=3$. For low-Tucker-rank linear functional inference, $\underline{\lambda} \in \{\overline{p}^{1/2}\overline{r}^{1/2}, \overline{p}^{3/4}\overline{r}^{1/2}, \overline{p}\overline{r}^{1/2}\}$. For general linear functional inference, $\underline{\lambda} \in \{\overline{p}^{3/4}\overline{r}^{1/2}, \overline{p}\overline{r}^{1/2}, \overline{p}^{5/4}\overline{r}^{1/2}\}$.

Theorems & Definitions (95)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 3.1: Main Theorem: Asymptotic Normality in Tensor Regression
• Remark 6
• Corollary 3.1: Asymptotic normality of estimated general linear functionals
• Theorem 3.2
• Corollary 3.2: Asymptotic normality of estimated low-Tucker-rank linear functionals