Matrix Completion with Deterministic Sampling: Theories and Methods

TL;DR

This paper investigates deterministic sampling for matrix completion, introducing new conditions that guarantee recoverability and establishing theoretical foundations beyond random sampling assumptions.

Contribution

It proposes the isomeric and relative well-conditionedness conditions, weaker than uniform sampling, and proves their necessity and sufficiency for matrix recovery.

Findings

Proves conditions for deterministic matrix completion.

Introduces the IsoDP method with unique behaviors.

Establishes theoretical guarantees beyond random sampling.

Abstract

In some significant applications such as data forecasting, the locations of missing entries cannot obey any non-degenerate distributions, questioning the validity of the prevalent assumption that the missing data is randomly chosen according to some probabilistic model. To break through the limits of random sampling, we explore in this paper the problem of real-valued matrix completion under the setup of deterministic sampling. We propose two conditions, isomeric condition and relative well-conditionedness, for guaranteeing an arbitrary matrix to be recoverable from a sampling of the matrix entries. It is provable that the proposed conditions are weaker than the assumption of uniform sampling and, most importantly, it is also provable that the isomeric condition is necessary for the completions of any partial matrices to be identifiable. Equipped with these new tools, we prove a collection of theorems for missing data recovery as well as convex/nonconvex matrix completion. Among other things, we study in detail a Schatten quasi-norm induced method termed isomeric dictionary pursuit (IsoDP), and we show that IsoDP exhibits some distinct behaviors absent in the traditional bilinear programs.