Statistical Inference in Tensor Completion: Optimal Uncertainty Quantification and Statistical-to-Computational Gaps

Abstract

This paper presents a simple yet efficient method for statistical inference of tensor linear forms using incomplete and noisy observations. Under the Tucker low-rank tensor model and the missing-at-random assumption, we utilize an appropriate initial estimate along with a debiasing technique followed by a one-step power iteration to construct an asymptotically normal test statistic. This method is suitable for various statistical inference tasks, including constructing confidence intervals, inference under heteroskedastic and sub-exponential noise, and simultaneous testing. We demonstrate that the estimator achieves the Cramér-Rao lower bound on Riemannian manifolds, indicating its optimality in uncertainty quantification. We comprehensively examine the statistical-to-computational gaps and investigate the impact of initialization on the minimal conditions regarding sample size and signal-to-noise ratio required for accurate inference. Our findings show that with independent initialization, statistically optimal sample sizes and signal-to-noise ratios are sufficient for accurate inference. Conversely, if only dependent initialization is available, computationally optimal sample sizes and signal-to-noise ratio conditions still guarantee asymptotic normality without the need for data-splitting. We present the phase transition between computational and statistical limits. Numerical simulation results align with the theoretical findings.

Figures (4)

• Figure 1: Phase transitions and statistical-to-computational gaps for inference on tensor linear forms. Region A: statistically impossible region; Region B: inference can be achieved by independent oracle initialization in Algorithm \ref{['alg:oracle-init']} with data-splitting; Region C: inference can be achieved by leave-one-out-type initialization cai2019nonconvexwang2023implicit without data-splitting; Region D: inference can be achieved by any warm initialization with guaranteed entywise error bound, with no data-splitting; Region E: inference can be achieved by computationally fast algorithms with no data-splitting.
• Figure 2: Histogram of normal approximation over 1000 independent trails with $\gamma=\{0.5,0.75,1,0.5\}$. The first 3 panels show results under homogeneous Guassian noises while the last panel shows results under heterogeneous Guassian noises.
• Figure 3: Histogram of normal approximation over 1000 independent trails with $\lambda_{\min} = 10 d^{\gamma}$ and $\gamma=\{0.75,1,1.25,0.75\}$. ${\mathbfcal{I}}$ is sparse 0-1 tensor with $\left\lvert\operatorname{supp}({\mathbfcal{I}})\right\rvert =10$. The last panel shows the results under heteroskedastic noises, where we set: $[\min_{\omega} [{\mathbfcal{S}}]_{\omega},\max_{\omega} [{\mathbfcal{S}}]_{\omega}] = [0.75,1.25]$.
• Figure 4: $\mathsf{AvgCov}$ of 0.95 and 0.9 level CIs over 100 independent trails. The error bars represent $\mathsf{AvgCov} \pm z_{0.1} \widehat{\sigma}$, showing the $80\%$ confidence intervals of the $\mathsf{AvgCov}$

Theorems & Definitions (54)

• Theorem 1: Lower bound on the variance of unbiased estimators
• Remark 1
• Theorem 2: Asymptotic normality with independent initialization
• Theorem 3: Inference with independent initialization
• Remark 2
• Corollary 1: Confidence interval
• Theorem 4: Inference with a leave-one-out initialization
• Theorem 5: Inference with an arbitrarily dependent initialization
• Theorem 6
• Proposition 1