RGNMR: A Gauss-Newton method for robust matrix completion with theoretical guarantees

Eilon Vaknin Laufer, Boaz Nadler

Keywords: Robust Matrix Completion, Outlier Detection, Gauss-Newton Method, Matrix Factorization, Robust PCA, Matrix Completion

TL;DR: A novel robust matrix completion method that overcomes limitations of existing schemes, accompanied by recovery guarantees

Abstract:

Recovering a low rank matrix from a subset of its entries,
some of which may be corrupted, is known as the robust matrix completion (RMC) problem.
Existing RMC methods have several limitations: they require a relatively large number of observed entries;
they may fail under overparametrization, when their assumed rank is higher than the correct one;
and many of them fail to recover even mildly ill-conditioned matrices.
In this paper we propose a novel RMC method, denoted $\texttt{RGNMR}$, which overcomes these limitations.
$\texttt{RGNMR}$ is a simple factorization-based iterative algorithm, which combines a Gauss–Newton linearization with removal of entries suspected to be outliers.
On the theoretical front, we prove that under suitable assumptions,
$\texttt{RGNMR}$ is guaranteed exact recovery of the underlying low rank matrix.
Our theoretical results improve upon the best currently known for factorization-based methods.
On the empirical front,
we show via several simulations
the advantages of $\texttt{RGNMR}$ over existing RMC methods, and in particular its ability to handle a small number of observed entries, overparameterization of the rank and ill-conditioned matrices.
In addition, we propose a novel scheme for estimating the number of corrupted entries.
This scheme may be used by other RMC methods that require as input the number of corrupted entries.

Primary Area: General machine learning (supervised, unsupervised, online, active, etc.)

Submission Number: 7362