Faster eigenvector computation via shift-and-invert preconditioning

Dan Garber, Elad Hazan, Chi Jin, Sham M. Kakade, Cameron Musco, Praneeth Netrapalli, Aaron Sidford

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We give faster algorithms and improved sample complexities for the fundamental problem of estimating the top eigenvector. Given an explicit matrix A € Rn×d, we show how to compute an e approximate top eigenvector of ATA in time O (jnnz(A) + • log l/ϵ). Here nnz(A) is the number of nonzeros in A, sr(A) is the stable rank, and gap is the relative eigengap. We also consider an online setting in which, given a stream of i.i.d. samples from a distribution V with covariance matrix E and a vector xq which is an O(gap) approximate top eigenvector for E, we show how to refine xo to an € approximation using O j samples from V. Here v(P) is a natural notion of variance. Combining our algorithm with previous work to initialize xo, we obtain improved sample complexities and runtimes under a variety of assumptions on V. We achieve our results via a robust analysis of the classic shift-and-invert preconditioning method. This technique lets us reduce eigenvector computation to approximately solving a scries of linear systems with fast stochastic gradient methods.

Original languageEnglish
Title of host publication33rd International Conference on Machine Learning, ICML 2016
EditorsKilian Q. Weinberger, Maria Florina Balcan
Pages3886-3894
Number of pages9
ISBN (Electronic)9781510829008
StatePublished - 2016
Externally publishedYes
Event33rd International Conference on Machine Learning, ICML 2016 - New York City, United States
Duration: 19 Jun 201624 Jun 2016

Publication series

Name33rd International Conference on Machine Learning, ICML 2016
Volume6

Conference

Conference33rd International Conference on Machine Learning, ICML 2016
Country/TerritoryUnited States
CityNew York City
Period19/06/1624/06/16

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Software
  • Computer Networks and Communications

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