Neural Comput - Active subspace: toward scalable low-rank learning.

Tópicos

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Resumo

We address the scalability issues in low-rank matrix learning problems. Usually these problems resort to solving nuclear norm regularized optimization problems (NNROPs), which often suffer from high computational complexities if based on existing solvers, especially in large-scale settings. Based on the fact that the optimal solution matrix to an NNROP is often low rank, we revisit the classic mechanism of low-rank matrix factorization, based on which we present an active subspace algorithm for efficiently solving NNROPs by transforming large-scale NNROPs into small-scale problems. The transformation is achieved by factorizing the large solution matrix into the product of a small orthonormal matrix (active subspace) and another small matrix. Although such a transformation generally leads to nonconvex problems, we show that a suboptimal solution can be found by the augmented Lagrange alternating direction method. For the robust PCA (RPCA) (Cand?s, Li, Ma, & Wright, 2009 ) problem, a typical example of NNROPs, theoretical results verify the suboptimality of the solution produced by our algorithm. For the general NNROPs, we empirically show that our algorithm significantly reduces the computational complexity without loss of optimality.

Resumo Limpo

address scalabl issu lowrank matrix learn problem usual problem resort solv nuclear norm regular optim problem nnrop often suffer high comput complex base exist solver especi largescal set base fact optim solut matrix nnrop often low rank revisit classic mechan lowrank matrix factor base present activ subspac algorithm effici solv nnrop transform largescal nnrop smallscal problem transform achiev factor larg solut matrix product small orthonorm matrix activ subspac anoth small matrix although transform general lead nonconvex problem show suboptim solut can found augment lagrang altern direct method robust pca rpca cand li ma wright problem typic exampl nnrop theoret result verifi suboptim solut produc algorithm general nnrop empir show algorithm signific reduc comput complex without loss optim

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