Low-Rank Matrix and Tensor Completion via Adaptive Sampling

TL;DR

This paper introduces adaptive sampling algorithms for low-rank matrix and tensor completion, achieving strong theoretical guarantees and efficient recovery even with high coherence or noise.

Contribution

It presents novel adaptive sampling algorithms that improve recovery guarantees for low-rank matrices and tensors, especially under high coherence conditions.

Findings

Exact recovery of rank-r matrices from O(n r^{3/2} log r) entries.

Recovery of order T tensors from O(n r^{T-1/2} T^2 log r) entries.

Robust noisy recovery with polylogarithmic sample complexity.

Abstract

We study low rank matrix and tensor completion and propose novel algorithms that employ adaptive sampling schemes to obtain strong performance guarantees. Our algorithms exploit adaptivity to identify entries that are highly informative for learning the column space of the matrix (tensor) and consequently, our results hold even when the row space is highly coherent, in contrast with previous analyses. In the absence of noise, we show that one can exactly recover a $n \times n$ matrix of rank $r$ from merely $\Omega(n r^{3/2}\log(r))$ matrix entries. We also show that one can recover an order $T$ tensor using $\Omega(n r^{T-1/2}T^2 \log(r))$ entries. For noisy recovery, our algorithm consistently estimates a low rank matrix corrupted with noise using $\Omega(n r^{3/2} \textrm{polylog}(n))$ entries. We complement our study with simulations that verify our theory and demonstrate the scalability of our algorithms.