Simple and Nearly-Optimal Sampling for Rank-1 Tensor Completion via Gauss-Jordan
Alejandro Gomez-Leos, Oscar López
TL;DR
This work studies the problem of completing a rank-1 tensor from uniformly sampled entries, a higher-order analogue of matrix completion. It introduces a simple Gauss-Jordan-based algorithm that reduces recovery to solving a pair of linear systems over $\mathbb{F}_2$ and $\mathbb{R}$ with a common coefficient matrix $\mathbf{A}$, whose rows enumerate all concatenations of base vectors. For constant $N$, the authors prove an upper bound of $m \lesssim (dN)^2 \log d$ samples with runtime $O(qN + m(dN)^2)$, and an information-theoretic lower bound of $\Omega(d \log dN)$ samples, with no dependence on incoherence $\mu$, highlighting a gap between rank-1 and general rank tensor completion. The results leverage a linear-algebraic reduction and a row-sampling argument, and raise open questions about tightening bounds and extending to larger $N$.
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 $N = Θ(1)$, 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 $Ω(d\log d)$ samples. By contrast, existing upper bounds on the sample complexity are at least as large as $d^{1.5} μ^{Ω(1)} \log^{Ω(1)} d$, where $μ$ can be $Θ(d)$ in the worst case. Prior work obtained these looser guarantees in higher rank versions of our problem, and tend to involve more complicated algorithms.