Low Permutation-rank Matrices: Structural Properties and Noisy Completion

TL;DR

This paper introduces the permutation-rank model for noisy matrix completion, providing theoretical minimax rates, analyzing an efficient algorithm, and exploring structural properties, offering a richer alternative to traditional low-rank assumptions.

Contribution

The paper proposes the permutation-rank model for matrix completion, establishes minimax estimation rates, and demonstrates that existing algorithms are effective under this new model.

Findings

Minimax rates for permutation-rank estimation match low-rank rates up to logarithmic factors.

Singular-value-thresholding algorithm is effective for permutation-rank matrices.

Structural results characterize permutation-rank decomposition uniqueness.

Abstract

We consider the problem of noisy matrix completion, in which the goal is to reconstruct a structured matrix whose entries are partially observed in noise. Standard approaches to this underdetermined inverse problem are based on assuming that the underlying matrix has low rank, or is well-approximated by a low rank matrix. In this paper, we propose a richer model based on what we term the "permutation-rank" of a matrix. We first describe how the classical non-negative rank model enforces restrictions that may be undesirable in practice, and how and these restrictions can be avoided by using the richer permutation-rank model. Second, we establish the minimax rates of estimation under the new permutation-based model, and prove that surprisingly, the minimax rates are equivalent up to logarithmic factors to those for estimation under the typical low rank model. Third, we analyze a computationally efficient singular-value-thresholding algorithm, known to be optimal for the low-rank setting, and show that it also simultaneously yields a consistent estimator for the low-permutation rank setting. Finally, we present various structural results characterizing the uniqueness of the permutation-rank decomposition, and characterizing convex approximations of the permutation-rank polytope.