Sample Complexity of Low-rank Tensor Recovery from Uniformly Random Entries

TL;DR

This paper establishes near-optimal bounds for the number of random entries needed to uniquely recover low-rank tensors of fixed order and rank, improving previous bounds significantly.

Contribution

It provides tight bounds on the sample complexity for tensor recovery from random entries, extending understanding to higher-order tensors and demonstrating the preservation of identifiability.

Findings

Recovery is possible with high probability from near-linear in n log n samples.

The bounds are tight up to a constant factor, improving previous polynomial bounds.

Projection of Segre variety preserves identifiability under specified conditions.

Abstract

We show that a generic tensor $T\in \mathbb{F}^{n\times n\times \dots\times n}$ of order $k$ and CP rank $d$ can be uniquely recovered from $n\log n+dn\log \log n +o(n\log \log n) $ uniformly random entries with high probability if $d$ and $k$ are constant and $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. The bound is tight up to the coefficient of the second leading term and improves on the existing $O(n^{\frac{k}{2}}{\rm polylog}(n))$ upper bound for order $k$ tensors. The bound is obtained by showing that the projection of the Segre variety to a random axis-parallel linear subspace preserves $d$-identifiability with high probability if the dimension of the subspace is $n\log n+dn\log \log n +o(n\log \log n) $ and $n$ is sufficiently large.