Efficient Proximal Mapping Computation for Low-Rank Inducing Norms

Christian Grussler () and Pontus Giselsson ()
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Christian Grussler: University of California, Berkeley
Pontus Giselsson: Lund University

Journal of Optimization Theory and Applications, 2022, vol. 192, issue 1, No 7, 168-194

Abstract: Abstract Low-rank inducing unitarily invariant norms have been introduced to convexify problems with a low-rank/sparsity constraint. The most well-known member of this family is the so-called nuclear norm. To solve optimization problems involving such norms with proximal splitting methods, efficient ways of evaluating the proximal mapping of the low-rank inducing norms are needed. This is known for the nuclear norm, but not for most other members of the low-rank inducing family. This work supplies a framework that reduces the proximal mapping evaluation into a nested binary search, in which each iteration requires the solution of a much simpler problem. The simpler problem can often be solved analytically as demonstrated for the so-called low-rank inducing Frobenius and spectral norms. The framework also allows to compute the proximal mapping of increasing convex functions composed with these norms as well as projections onto their epigraphs.

Keywords: Low-rank optimization; Low-rank inducing norms; Proximal splitting; Regularization; Matrix completion; 90C06; 90C25; 90C26; 15A83; 90C30; 90C59; 90C90 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-021-01956-2

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