TY - JOUR

T1 - Scale invariant power iteration

AU - Kim, Cheolmin

AU - Kim, Youngseok

AU - Klabjan, Diego

N1 - Publisher Copyright:
Copyright © 2019, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2019/5/23

Y1 - 2019/5/23

N2 - Power iteration has been generalized to solve many interesting problems in machine learning and statistics. Despite its striking success, theoretical understanding of when and how such an algorithm enjoys good convergence property is limited. In this work, we introduce a new class of optimization problems called scale invariant problems and prove that they can be efficiently solved by scale invariant power iteration (SCI-PI) with a generalized convergence guarantee of power iteration. By deriving that a stationary point is an eigenvector of the Hessian evaluated at the point, we show that scale invariant problems indeed resemble the leading eigenvector problem near a local optimum. Also, based on a novel reformulation, we geometrically derive SCI-PI which has a general form of power iteration. The convergence analysis shows that SCI-PI attains local linear convergence with a rate being proportional to the top two eigenvalues of the Hessian at the optimum. Moreover, we discuss some extended settings of scale invariant problems and provide similar convergence results for them. In numerical experiments, we introduce applications to independent component analysis, Gaussian mixtures, and non-negative matrix factorization. Experimental results demonstrate that SCI-PI is competitive to state-of-the-art benchmark algorithms and often yield better solutions.

AB - Power iteration has been generalized to solve many interesting problems in machine learning and statistics. Despite its striking success, theoretical understanding of when and how such an algorithm enjoys good convergence property is limited. In this work, we introduce a new class of optimization problems called scale invariant problems and prove that they can be efficiently solved by scale invariant power iteration (SCI-PI) with a generalized convergence guarantee of power iteration. By deriving that a stationary point is an eigenvector of the Hessian evaluated at the point, we show that scale invariant problems indeed resemble the leading eigenvector problem near a local optimum. Also, based on a novel reformulation, we geometrically derive SCI-PI which has a general form of power iteration. The convergence analysis shows that SCI-PI attains local linear convergence with a rate being proportional to the top two eigenvalues of the Hessian at the optimum. Moreover, we discuss some extended settings of scale invariant problems and provide similar convergence results for them. In numerical experiments, we introduce applications to independent component analysis, Gaussian mixtures, and non-negative matrix factorization. Experimental results demonstrate that SCI-PI is competitive to state-of-the-art benchmark algorithms and often yield better solutions.

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M3 - Article

AN - SCOPUS:85093447022

JO - Free Radical Biology and Medicine

JF - Free Radical Biology and Medicine

SN - 0891-5849

ER -