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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
• Solve overdetermined linear system
where , , .• Goal: compute .
– Least Square Regression (LSR)
Closed-form solution
• Closed-form solution known:
– : Moore-Penrose pseudo-inverse of .– If the SVD of is where , then .
• Classical complexity: • Prohibitively slow for big matrices .
Relaxations
• Relaxation: – Approximate: output .– Important special case: Sparse and low-rank :
*1,2, where • # non-zero entries in each row/column.• .
• Quantum speedup? Even writing down the solution takes linear time.
*1. K. Clarkson, D. Woodruff. STOC, 2013.*2. J. Nelson, H. Nguyen. FOCS, 2013.
Quantum sketch
• Similar issue as solving linear system for full-rank . – Closed-form solution:
• [HHL09]*1 Output in time
• Condition number , where are ’s singular values.
• : sparsity. • proportional
*1. A. Harrow, A. Hassidim, S. Lloyd, PRL, 2009.
Controversy
• Useless? Can’t read out each solution variable ’s.
• Useful? As intermediate steps, e.g. when some global info of is needed. – can be obtained from by SWAP test.
• Classically also ? Impossible unless
LSR results
• Back to overdetermined system: .• [WBL12]*1: Output in time .• Ours:
– Same approx. in time – Simpler algorithm. – Can also estimate , which is used for, e.g.
computing .– Extensions: Ridge Regression, Truncated
SVD *1. N. Wiebe, D. Braun, S. Lloyd, PRL, 2012.
Our algorithm for LSR
• Input: Hermition ,. Assume with , and the rest ’s are 0.– Non-Hermition reduces to Hermition.
• Output: w/ , and .• Note: Write as , then the desirable output
is .
Algorithm
•
where
// attach , rotate if
// “select” component
, which is just .
Tool: Phase Estimation quantum algorithm. Output eigenvalue for a given eigenvector.
Extension 1: Ridge regression
• For ill-conditioned (i.e. large ) input?• Two classical solutions.• Ridge regression: .
– Closed-form solution: – Previous algorithms: ,
• for sparse and low rank.
• Ours: , for .
Part II. statistic leverage scores
• has SVD . The -th leverage score – : the -th row of .
• Matrix coherence: .• Leverage score measures the importance
of row .– A well-studied measure.– Very useful in large scale data analysis,
matrix algorithms, outlier detection, low-rank matrix approximation, etc. *1
*1. M. Mahoney, Randomized Algorithms for Matrices and Data, Foundations & Trends in Machine Learning, 2010.
Computing leverage scores
• Classical algo.*1 finding all : .• No better algorithm for finding • Our quantum algorithms for
– finding each : .– finding all : .– finding : .
*1. P. Drineas, M. Magdon-Ismail, M. Mahoney, D. Woodruff. J. MLR, 2012.
Algorithm
• where // rotate to if .
• Estimate the prob of observing 1 when measuring the last qubit.
• , the target.