A quantum-inspired algorithm for approximating statistical leverage scores
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Suppose a matrix A ∈ℝ^m × n of rank k with singular value decomposition A = U_AΣ_A V_A^T, where U_A∈ℝ^m × k, V_A∈ℝ^n × k are orthonormal and Σ_A∈ℝ^k × k is a diagonal matrix. The statistical leverage scores of a matrix A are the squared row-norms defined by ℓ_i = (U_A)_i,:_2^2, where i ∈ [m], and the matrix coherence is the largest statistical leverage score. These quantities play an important role in machine learning algorithms such as matrix completion and Nyström-based low rank matrix approximation as well as large-scale statistical data analysis applications. The best known classical algorithm to approximate these values runs in time O((mn + n^3) log m) in [P. Drineas, M. Magdon-Ismail, M. W. Mahoney and D. P. Woodruff. Fast approximation of matrix coherence and statistical leverage. J. Mach. Learn. Res., (2012)13: 3475-3506]. In this work, inspired by recent development on dequantization techniques, we propose a fast novel classical algorithm for approximating the statistical leverage scores. Our novel algorithm has query and time complexity O( poly(k, κ, 1/ϵ, 1/δ, log(mn)) ), where κ is the condition number of A, and δ is the failure probability.
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