Alignment and matching tests for high-dimensional tensor signals via tensor contraction

TL;DR

The paper develops a rigorous spectral-method framework for testing high-dimensional tensor signals via a tensor contraction to a data matrix with correlated entries. It establishes the limiting spectral distribution through a vector Dyson equation and proves a central limit theorem for linear spectral statistics, enabling precise, analytically tractable test statistics for tensor signal alignment and matching. The proposed tests quantify signal alignment through a drift-augmented normalization and extend to general d-fold tensors, with plug-in estimators for cumulants allowing practical implementation. Numerical experiments and real video data analysis demonstrate accurate finite-sample performance, robust power below phase transitions, and clear separation between matching and nonmatching signal structures in real data.

Abstract

We consider two hypothesis testing problems for low-rank and high-dimensional tensor signals, namely the tensor signal alignment and tensor signal matching problems. These problems are challenging due to the high dimension of tensors and the lack of suitable test statistics. By exploiting a recent tensor contraction method, we propose and validate relevant test statistics using eigenvalues of a data matrix resulting from the tensor contraction. The matrix entries exhibit long-range dependence, which makes the analysis of the matrix challenging, involved, and distinct from standard random matrix theory. Our approach provides a novel framework for addressing hypothesis testing problems in the context of high-dimensional tensor signals.

Figures (6)

• Figure 1: QQ plots of $G_N(f)$ from 100 independent repetitions.
• Figure 2: Power plots of $\tilde{\mathcal{T}}_N^{(3)}(\boldsymbol{x}^{(1)},\boldsymbol{x}^{(2)},\boldsymbol{x}^{(3)})$ under different $\beta$'s and types of noises $\boldsymbol{X}$ and vectors $\boldsymbol{x}^{(i)}$, where the dashed red line is the significance level $\alpha=0.05$ and the dashed blue line is the threshold of phase transition.
• Figure 3: Power plots of $\tilde{\mathcal{T}}_N^{(3)}(\hat{\boldsymbol{x}}^{(1)},\hat{\boldsymbol{x}}^{(2)},\hat{\boldsymbol{x}}^{(3)})$ under different $\beta_0,\beta_1$ and types of noises $\boldsymbol{X}$ and vectors $\boldsymbol{x}^{(i)}$. "Known" denotes the empirical power of $\tilde{\mathcal{T}}_N^{(3)}(\boldsymbol{x}^{(1)},\boldsymbol{x}^{(2)},\boldsymbol{x}^{(3)})$, while "Beta0=$a$" represents the empirical power of $\tilde{\mathcal{T}}_N^{(3)}(\hat{\boldsymbol{x}}^{(1)},\hat{\boldsymbol{x}}^{(2)},\hat{\boldsymbol{x}}^{(3)})$ when $\beta_0=a$, $a=2,2.5,3$. The dashed red line and blue line indicate the significance level $\alpha=0.05$ and the threshold of phase transition $\beta_s=2/\sqrt 3=1.1547$, respectively.
• Figure 4: Histogram of acceptance rates from Table \ref{['Main of Tab of real data analysis']}, confirming that videos of different actions show distinct signal structures with our test.
• Figure 5: Power plots of $\tilde{\mathcal{T}}_N^{(3)}(\boldsymbol{x}^{(1)},\boldsymbol{x}^{(2)},\boldsymbol{x}^{(3)})$ under different $\beta$'s and types of noises $\boldsymbol{X}$ and vectors $\boldsymbol{x}^{(i)}$, where the dashed red line is the significance level $\alpha=0.05$ and the dashed blue line is the threshold of phase transition.

Theorems & Definitions (112)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3: Entrywise law
• Remark 2.1
• Remark 2.2
• Remark 3.1
• Theorem 3.1
• Remark 3.2
• Proposition 3.1: Mean function $\mu_N^{(3)}(z)$ for $d=3$
• Proposition 3.2: Covariance function $\mathcal{C}_N^{(3)}(z_1,z_2)$ for $d=3$