Tensor Sandwich: Tensor Completion for Low CP-Rank Tensors via Adaptive Random Sampling

TL;DR

This paper introduces an adaptive tensor completion method that efficiently recovers low CP-rank tensors by strategically sampling slices and combining matrix completion with a noise-robust algorithm, achieving high accuracy with fewer samples.

Contribution

It presents a novel adaptive sampling strategy combined with matrix completion and a modified Jennrich's algorithm for efficient tensor completion.

Findings

Achieves tensor completion with O(nr log^2 r) samples under mild assumptions.

Works effectively as a low-rank approximation method in noisy settings.

Empirical results confirm practical effectiveness of the proposed approach.

Abstract

We propose an adaptive and provably accurate tensor completion approach based on combining matrix completion techniques (see, e.g., arXiv:0805.4471, arXiv:1407.3619, arXiv:1306.2979) for a small number of slices with a modified noise robust version of Jennrich's algorithm. In the simplest case, this leads to a sampling strategy that more densely samples two outer slices (the bread), and then more sparsely samples additional inner slices (the bbq-braised tofu) for the final completion. Under mild assumptions on the factor matrices, the proposed algorithm completes an $n \times n \times n$ tensor with CP-rank $r$ with high probability while using at most $\mathcal{O}(nr\log^2 r)$ adaptively chosen samples. Empirical experiments further verify that the proposed approach works well in practice, including as a low-rank approximation method in the presence of additive noise.