Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, May 02, 2017, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Kenneth Clarkson (IBM Research Almaden)

Low-rank PSD Approximation in Input-Sparsity Time

A number of matrices that arise in machine learning and data analysis are symmetric positive semidefinite (PSD), including covariance matrices, kernel matrices, Laplacian matrices, random dot product graph models, and others. A common task related to such matrices is to approximate them with a low-rank matrix, for efficiency or statistical inference; spectral clustering, kernel PCA, manifold learning, and Gaussian process regression can all involve this task. Given a square n by n matrix A, target rank k, and error parameter epsilon, we show how to find a PSD matrix B of rank k, such that the Frobenius norm of A-B is within (1+epsilon) of best possible; our algorithm needs O(nnz(A) + n*poly(k/epsilon)) time to do this, where nnz(A) is the number of nonzero entries of A. We also show how to find such a rank-k matrix PSD B of the form CUC^T, where the O(k/eps) columns of C are a subset of those of A, with a similar runtime. Joint work with David Woodruff.

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