Near-optimal sample complexity for convex tensor completion

TL;DR

This paper introduces a new tensor completion method using atomic and max-qnorms that achieves near-optimal sample complexity, significantly improving over previous nuclear-norm approaches and matching theoretical lower bounds.

Contribution

It proposes atomic-norm and max-qnorm based tensor completion techniques that attain nearly minimax optimal sample complexity, advancing the theoretical understanding and practical performance.

Findings

Achieves $O(dN)$ sample complexity for tensor completion.

Max-qnorm constrained estimation improves recovery over existing methods.

Numerical results demonstrate superior performance of the proposed methods.

Abstract

We analyze low rank tensor completion (TC) using noisy measurements of a subset of the tensor. Assuming a rank-$r$, order-$d$, $N \times N \times \cdots \times N$ tensor where $r=O(1)$, the best sampling complexity that was achieved is $O(N^{\frac{d}{2}})$, which is obtained by solving a tensor nuclear-norm minimization problem. However, this bound is significantly larger than the number of free variables in a low rank tensor which is $O(dN)$. In this paper, we show that by using an atomic-norm whose atoms are rank-$1$ sign tensors, one can obtain a sample complexity of $O(dN)$. Moreover, we generalize the matrix max-norm definition to tensors, which results in a max-quasi-norm (max-qnorm) whose unit ball has small Rademacher complexity. We prove that solving a constrained least squares estimation using either the convex atomic-norm or the nonconvex max-qnorm results in optimal sample complexity for the problem of low-rank tensor completion. Furthermore, we show that these bounds are nearly minimax rate-optimal. We also provide promising numerical results for max-qnorm constrained tensor completion, showing improved recovery results compared to matricization and alternating least squares.