A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples

Christian Kümmerle; Claudio Mayrink Verdun

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arXiv · 2106.02119

A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples

Christian Kümmerle·Claudio Mayrink Verdun

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Abstract

We propose an iterative algorithm for low-rank matrix completion that can be interpreted as an iteratively reweighted least squares (IRLS) algorithm, a saddle-escaping smoothing Newton method or a variable metric proximal gradient method applied to a non-convex rank surrogate. It combines the favorable data-efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. We establish the first local convergence guarantee from a minimal number of samples for that class of algorithms, showing that the method attains a local quadratic convergence rate. Furthermore, we show that the linear systems to be solved are well-conditioned even for very ill-conditioned ground truth matrices. We provide extensive experiments, indicating that unlike many state-of-the-art approaches, our method is able to complete very ill-conditioned matrices with a condition number of up to $10^{10}$ from few samples, while being competitive in its scalability.

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BibTeX RIS

Christian Kümmerle, Claudio Mayrink Verdun. 2021-06-03. A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples. https://arxiv.org/abs/2106.02119

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