TY - GEN

T1 - Faster eigenvector computation via shift-and-invert preconditioning

AU - Garber, Dan

AU - Hazan, Elad

AU - Jin, Chi

AU - Kakade, Sham M.

AU - Musco, Cameron

AU - Netrapalli, Praneeth

AU - Sidford, Aaron

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

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M3 - Conference contribution

AN - SCOPUS:84998611056

T3 - 33rd International Conference on Machine Learning, ICML 2016

SP - 3886

EP - 3894

BT - 33rd International Conference on Machine Learning, ICML 2016

A2 - Weinberger, Kilian Q.

A2 - Balcan, Maria Florina

PB - International Machine Learning Society (IMLS)

T2 - 33rd International Conference on Machine Learning, ICML 2016

Y2 - 19 June 2016 through 24 June 2016

ER -