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

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

URL: http://arxiv.org/abs/2408.05431v2
Date: Mon, 19 Aug 2024 22:30:20 GMT
Title: Simple and Nearly-Optimal Sampling for Rank-1 Tensor Completion via Gauss-Jordan
Authors: Alejandro Gomez-Leos, Oscar López,
Abstract summary: We revisit the sample and computational complexity of completing a rank-1 tensor in $otimes_i=1N mathbbRd$. We present a characterization of the problem which admits an algorithm amounting to Gauss-Jordan on a pair of random linear systems.
Score: 49.1574468325115
License: http://creativecommons.org/licenses/by/4.0/
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.

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