A quantum-inspired algorithm for approximating statistical leverage scores

Abstract

Suppose a matrix A ∈ R m×n of rank k with singular value decomposition A = U A Σ A V T A , 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 ǫ , 1 δ , log(mn) , where κ is the condition number of A, and δ is the failure probability.

An improved quantum algorithm for low-rank rigid linear regressions with vector solution outputs

“…In [35], Liu and Zhang proposed a quantum algorithm for approximating all leverage scores by using quantum phase estimation. Recently, in [57], Zuo and Xiang proposed a quantum-inspired classical algorithm for approximating leverage scores.…”

pylspack: Parallel algorithms and data structures for sketching, column subset selection, regression and leverage scores

“…Such aspects were mentioned, but not elaborated, in our recent companion paper [78]. We here aim to contribute in this direction, by providing a detailed analysis for a set of speci c operations and sketching transforms which are critical for various important problems in RandNLA and beyond, for example leverage scores computation [36,25,67,4,58,29,28,88,78], least squares regression [75,67,85,72,35,8,64,70], column subset selection [23,17,10,24,77,78], and rank computation [76,23,22,15,62]. Speci cally, given a talland-skinny matrix 𝐴 ∈ R 𝑛×𝑑 with 𝑛 𝑑, we present parallel algorithms, data structures and a numerical library for the following set of operations, all of which can be seen as special cases of sparse matrix multiplication:…”

Quantum-inspired algorithm for truncated total least squares solution

“…Compared with the quantum recommendation systems [50], Tang [51] proposes a quantum-inspired classical algorithm for recommendation systems within logarithmic time by using the efficient low-rank approximation techniques of Frieze, Kannan and Vempala (FKV) algorithm [52]. Motivated by the dequantizing techniques, other papers are also proposed to deal with some low-rank matrix operations, such as matrix inversion, singular value transformation, non-negative matrix factorization, support vector machine, general minimum conical hull problems, principal component analysis, canonical correlation analysis, statistical leverage scores [53][54][55][56][57][58][59][60][61]. Dequantizing techniques in those algorithms involve two technologies, the Monte-Carlo singular value decomposition and sampling techniques, which could efficiently simulate some special operations on low-rank matrices.…”