A Multilevel Monte Carlo Estimator for Matrix Multiplication

Inspired by the latest developments in multilevel Monte Carlo (MLMC) methods and randomised sketching for linear algebra problems we propose a MLMC estimator for real-time processing of matrix structured random data. Our algorithm is particularly effective in handling high-dimensional inner products and matrix multiplication, in applications of image analysis and large-scale supervised learning.

[1]  Petros Drineas,et al.  Fast Monte Carlo Algorithms for Matrices I: Approximating Matrix Multiplication , 2006, SIAM J. Comput..

[2]  Ilse C. F. Ipsen,et al.  Randomized Approximation of the Gram Matrix: Exact Computation and Probabilistic Bounds , 2015, SIAM J. Matrix Anal. Appl..

[3]  Elisabeth Ullmann,et al.  Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients , 2012, Numerische Mathematik.

[4]  A. Kebaier,et al.  Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing , 2005, math/0602529.

[5]  A. Beskos,et al.  Multilevel sequential Monte Carlo samplers , 2015, 1503.07259.

[6]  Harish Karnick,et al.  Random Feature Maps for Dot Product Kernels , 2012, AISTATS.

[7]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[8]  Benjamin Recht,et al.  Random Features for Large-Scale Kernel Machines , 2007, NIPS.

[9]  Alexey Chernov,et al.  Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems , 2015, Numerische Mathematik.

[10]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[11]  Ilse C. F. Ipsen,et al.  Importance Sampling for a Monte Carlo Matrix Multiplication Algorithm, with Application to Information Retrieval , 2011, SIAM J. Sci. Comput..

[12]  W. Schachermayer,et al.  Multilevel quasi-Monte Carlo path simulation , 2009 .

[13]  Michael B. Giles,et al.  Multilevel quasi-Monte Carlo path simulation , 2009 .

[14]  Yue Wu.  A Note on Random Sampling for Matrix Multiplication , 2018, ArXiv.