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

• URL: http://arxiv.org/abs/2410.11225v2
• Date: Fri, 01 Nov 2024 15:51:56 GMT
• Title: Statistical Inference in Tensor Completion: Optimal Uncertainty Quantification and Statistical-to-Computational Gaps
• Authors: Wanteng Ma, Dong Xia,
• Abstract summary: This paper presents a simple yet efficient method for statistical inference of tensor linear forms using incomplete and noisy observations. It is suitable for various statistical inference tasks, including constructing confidence intervals, inference under heteroskedastic and sub-exponential noise, and simultaneous testing.
• Score: 7.174572371800217
• License:
• 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 Cram\'er-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.

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