Simple and Nearly-Optimal Sampling for Rank-1 Tensor Completion via Gauss-Jordan
This work provides a nearly-optimal solution for tensor completion in a specific setting, which is incremental compared to prior methods with looser bounds.
The paper tackles the problem of completing a rank-1 tensor from uniformly sampled entries, presenting an algorithm based on Gauss-Jordan elimination that uses O(d^2 log d) samples and runs in O(md^2) time, while proving a lower bound of Ω(d log d) samples.
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.