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The Algebraic Combinatorial Approach for Low-Rank Matrix Completion

By Franz J. Király, Louis Theran and Ryota Tomioka

Abstract

We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the treatment of single entries in a closed theoretical and practical framework. More specifically, apart from introducing an algebraic combinatorial theory of low-rank matrix completion, we present probability-one algorithms to decide whether a particular entry of the matrix can be completed. We also describe methods to complete that entry from a few others, and to estimate the error which is incurred by any method completing that entry. Furthermore, we show how known results on matrix completion and their sampling assumptions can be related to our new perspective and interpreted in terms of a completability phase transition.Comment: 37 pages, with an appendix by Takeaki Un

Topics: Computer Science - Machine Learning, Mathematics - Algebraic Geometry, Mathematics - Combinatorics, Mathematics - Numerical Analysis, Statistics - Machine Learning
Year: 2014
OAI identifier: oai:arXiv.org:1211.4116

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