Rectangular maximum volume and projective volume search algorithms

New methods for finding submatrices of (locally) maximal volume and large projective volume are proposed and studied. Detailed analysis is also carried out for existing methods. The effectiveness of the new methods is shown in the construction of cross approximations, and estimates are also proved in the case of their application for the search for a strongly nondegenerate submatrix. Much attention is also paid to the choice of the starting submatrix.

[1]  Ivan V. Oseledets,et al.  Rectangular maximum-volume submatrices and their applications , 2015, ArXiv.

[2]  Christos Boutsidis,et al.  Near-Optimal Column-Based Matrix Reconstruction , 2014, SIAM J. Comput..

[3]  Christos Boutsidis,et al.  Faster Subset Selection for Matrices and Applications , 2011, SIAM J. Matrix Anal. Appl..

[4]  Gene H. Golub,et al.  Matrix computations , 1983 .

[5]  S. Goreinov,et al.  The maximum-volume concept in approximation by low-rank matrices , 2001 .

[6]  Santosh S. Vempala,et al.  Matrix approximation and projective clustering via volume sampling , 2006, SODA '06.

[7]  Christos Boutsidis,et al.  Optimal CUR matrix decompositions , 2014, STOC.

[8]  A. I. Osinsky,et al.  On the Existence of a Nearly Optimal Skeleton Approximation of a Matrix in the Frobenius Norm , 2018 .

[9]  S. Goreinov,et al.  A Theory of Pseudoskeleton Approximations , 1997 .

[10]  Santosh S. Vempala,et al.  Adaptive Sampling and Fast Low-Rank Matrix Approximation , 2006, APPROX-RANDOM.

[11]  Ming Gu,et al.  Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization , 1996, SIAM J. Sci. Comput..

[12]  Rectangular submatrices of maximum volume and their computation , 2015 .

[13]  Eugene E. Tyrtyshnikov,et al.  A fast numerical method for the Cauchy problem for the Smoluchowski equation , 2015, J. Comput. Phys..

[14]  Eugene E. Tyrtyshnikov,et al.  Quasioptimality of skeleton approximation of a matrix in the Chebyshev norm , 2011 .

[15]  Mario Bebendorf,et al.  Approximation of boundary element matrices , 2000, Numerische Mathematik.

[16]  Alexander Osinsky,et al.  Global Optimization Algorithms Using Tensor Trains , 2017, LSSC.