Yang Liu Shengyu Zhang The Chinese University of Hong Kong Fast quantum algorithms for Least Squares Regression and Statistic Leverage Scores.

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Presentation transcript:

Yang Liu Shengyu Zhang The Chinese University of Hong Kong Fast quantum algorithms for Least Squares Regression and Statistic Leverage Scores

Part I. Linear regression –Output a “quantum sketch” of solution. Part II. Computing leverage scores and matrix coherence. –Output the target numbers.

Part I: Linear regression

Closed-form solution

Relaxations *1. K. Clarkson, D. Woodruff. STOC, *2. J. Nelson, H. Nguyen. FOCS, 2013.

Quantum sketch *1. *1. A. Harrow, A. Hassidim, S. Lloyd, PRL, 2009.

Controversy

LSR results *1. *1. N. Wiebe, D. Braun, S. Lloyd, PRL, 2012.

Our algorithm for LSR

Algorithm Tool: Phase Estimation quantum algorithm. Output eigenvalue for a given eigenvector.

Extension 1: Ridge regression

Extension 2: Truncated SVD

Part II. statistic leverage scores *1. M. Mahoney, Randomized Algorithms for Matrices and Data, Foundations & Trends in Machine Learning, 2010.

Computing leverage scores *1. P. Drineas, M. Magdon-Ismail, M. Mahoney, D. Woodruff. J. MLR, 2012.

Algorithm for LSR

Algorithm

Summary We give efficient quantum algorithms for two canonical problems on sparse inputs –Least squares regression –Statistical leverage score The problems are linear algebraic, not group/number/polynomial theoretic