Low-Rank Matrix Recovery using Unconstrained Smoothed-Lq Minimization
A low-rank matrix can be recovered from a small number of its linear measurements. As a special case, the matrix completion problem aims to recover the matrix from a subset of its entries. Such problems share many common features with those recovering sparse vectors. In this paper, we extend nonconvex Lq minimization and iteratively reweighted algorithms from recovering sparse vectors to recovering low-rank matrices. Unlike most existing work, this work focuses on unconstrained Lq minimization, for which we show a few advantages on noisy measurements and/or approximately low-rank matrices. Based on results in Daubechies-DeVore-Fornasier-Güntürk '2010 for constrained Lq minimization, we start with a preliminary yet novel analysis for unconstrained Lq minimization for sparse vectors, which includes convergence, error bound, and local convergence rates. Then, the algorithm and analysis is extended to the recovery of low-rank matrices. The algorithm has been compared to some existing state-of-the-arts and shows superior performance on recovering low-rank matrices with fast-decaying singular values from incomplete measurements.
Citable link to this pagehttps://hdl.handle.net/1911/102186
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