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

arXiv: https://arxiv.org/abs/2410.11225 · HTML: https://mlsanity.com/paper/2410.11225/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 37 references that are in the map.
- First posted on arXiv 15 Oct 2024.

## Builds on (most cited first, 14 of 14)

- [Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution](https://mlsanity.com/paper/1711.10467/) (2017)
- [Inference and Uncertainty Quantification for Noisy Matrix Completion](https://mlsanity.com/paper/1906.04159/) (2019)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [On Tensor Completion via Nuclear Norm Minimization](https://mlsanity.com/paper/1405.1773/) (2014)
- [Community Detection on Mixture Multi-layer Networks via Regularized Tensor Decomposition](https://mlsanity.com/paper/2002.04457/) (2020)
- [Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization](https://mlsanity.com/paper/2103.08895/) (2021)
- [Factor Models for High-Dimensional Tensor Time Series](https://mlsanity.com/paper/1905.07530/) (2019)
- [Statistical Inferences of Linear Forms for Noisy Matrix Completion](https://mlsanity.com/paper/1909.00116/) (2019)
- [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022)
- [Latent Space Model for Higher-order Networks and Generalized Tensor Decomposition](https://mlsanity.com/paper/2106.16042/) (2021)
- [Noisy Tensor Completion via Low-rank Tensor Ring](https://mlsanity.com/paper/2203.08857/) (2022)
- [Inference for Low-rank Tensors -- No Need to Debias](https://mlsanity.com/paper/2012.14844/) (2020)
- [Matrix Completion When Missing Is Not at Random and Its Applications in Causal Panel Data Models](https://mlsanity.com/paper/2308.02364/) (2023)
- [Inference for Low-Rank Models](https://mlsanity.com/paper/2107.02602/) (2021)

## Built on by (most cited first, 1 of 1)

- [LLM Evaluation as Tensor Completion: Low Rank Structure and Semiparametric Efficiency](https://mlsanity.com/paper/2604.05460/) (2026)

## Abstract (arXiv)

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.