Simple and Nearly-Optimal Sampling for Rank-1 Tensor Completion via Gauss-Jordan

TL;DR

This paper introduces a simple, nearly-optimal algorithm based on Gauss-Jordan elimination for completing rank-1 tensors from sampled entries, achieving lower sample complexity and computational efficiency than previous methods.

Contribution

The paper presents a novel, efficient Gauss-Jordan-based algorithm for rank-1 tensor completion with improved sample complexity bounds and simplicity over prior approaches.

Findings

Uses no more than O(d^2 log d) samples for N=Θ(1)

Runs in O(md^2) time, where m is the number of samples

Any algorithm requires at least Ω(d log d) samples

Abstract

We revisit the sample and computational complexity of completing a rank-1 tensor in $\otimes_{i=1}^{N} \mathbb{R}^{d}$, given a uniformly sampled subset of its entries. We present a characterization of the problem (i.e. nonzero entries) which admits an algorithm amounting to Gauss-Jordan on a pair of random linear systems. For example, when $N = \Theta(1)$, we prove it uses no more than $m = O(d^2 \log d)$ samples and runs in $O(md^2)$ time. Moreover, we show any algorithm requires $\Omega(d\log d)$ samples. By contrast, existing upper bounds on the sample complexity are at least as large as $d^{1.5} \mu^{\Omega(1)} \log^{\Omega(1)} d$, where $\mu$ can be $\Theta(d)$ in the worst case. Prior work obtained these looser guarantees in higher rank versions of our problem, and tend to involve more complicated algorithms.