Scale Invariant Power Iteration

TL;DR

This paper introduces scale invariant power iteration (SCI-PI), a new algorithm for solving a class of scale invariant problems with provable local linear convergence, and demonstrates its effectiveness in machine learning applications.

Contribution

The paper defines scale invariant problems, develops SCI-PI with convergence guarantees, and shows its competitive performance in practical machine learning tasks.

Findings

SCI-PI achieves local linear convergence proportional to top eigenvalues.

SCI-PI outperforms state-of-the-art algorithms in experiments.

Applications include ICA, Gaussian mixtures, and NMF with promising results.

Abstract

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.