Matrix completion with column manipulation: Near-optimal sample-robustness-rank tradeoffs

TL;DR

This paper introduces an efficient matrix completion algorithm resilient to column corruption, achieving near-optimal tradeoffs between sample size, robustness, and matrix rank, with broad applications including robust collaborative filtering.

Contribution

It presents a novel convex optimization approach combining trimming and nuclear norm minimization that tolerates arbitrary column corruptions without prior assumptions.

Findings

Algorithm guarantees matrix recovery with a vanishing fraction of observed entries.

Theoretical bounds show near-optimal robustness against corrupted columns.

Results characterize fundamental tradeoffs between sample size, corruption level, and matrix rank.

Abstract

This paper considers the problem of matrix completion when some number of the columns are completely and arbitrarily corrupted, potentially by a malicious adversary. It is well-known that standard algorithms for matrix completion can return arbitrarily poor results, if even a single column is corrupted. One direct application comes from robust collaborative filtering. Here, some number of users are so-called manipulators who try to skew the predictions of the algorithm by calibrating their inputs to the system. In this paper, we develop an efficient algorithm for this problem based on a combination of a trimming procedure and a convex program that minimizes the nuclear norm and the $\ell_{1,2}$ norm. Our theoretical results show that given a vanishing fraction of observed entries, it is nevertheless possible to complete the underlying matrix even when the number of corrupted columns grows. Significantly, our results hold without any assumptions on the locations or values of the observed entries of the manipulated columns. Moreover, we show by an information-theoretic argument that our guarantees are nearly optimal in terms of the fraction of sampled entries on the authentic columns, the fraction of corrupted columns, and the rank of the underlying matrix. Our results therefore sharply characterize the tradeoffs between sample, robustness and rank in matrix completion.