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Recovering Simultaneously Structured Data via Non-Convex Iteratively Reweighted Least Squares

Christian Kümmerle · Christian Kümmerle · Johannes Maly

Great Hall & Hall B1+B2 (level 1) #1121
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[ Paper [ Slides [ Poster [ OpenReview
Tue 12 Dec 3:15 p.m. PST — 5:15 p.m. PST


We propose a new algorithm for the problem of recovering data that adheres to multiple, heterogenous low-dimensional structures from linear observations. Focussing on data matrices that are simultaneously row-sparse and low-rank, we propose and analyze an iteratively reweighted least squares (IRLS) algorithm that is able to leverage both structures. In particular, it optimizes a combination of non-convex surrogates for row-sparsity and rank, a balancing of which is built into the algorithm. We prove locally quadratic convergence of the iterates to a simultaneously structured data matrix in a regime of minimal sample complexity (up to constants and a logarithmic factor), which is known to be impossible for a combination of convex surrogates. In experiments, we show that the IRLS method exhibits favorable empirical convergence, identifying simultaneously row-sparse and low-rank matrices from fewer measurements than state-of-the-art methods.

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