Rectangular maximumvolume submatrices and their applications
Abstract
We introduce a definition of the volume for a general rectangular matrix, which for square matrices is equivalent to the absolute value of the determinant. We generalize results for square maximumvolume submatrices to the case of rectangular maximalvolume submatrices, show connection of the rectangular volume with optimal experimental design and provide estimates for the growth of the coefficients and approximation error in spectral and Chebyshev norms. Three promising applications of such submatrices are presented: recommender systems, finding maximal elements in lowrank matrices and preconditioning of overdetermined linear systems. The code is available online.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.07838
 Bibcode:
 2015arXiv150207838M
 Keywords:

 Mathematics  Numerical Analysis;
 Computer Science  Numerical Analysis;
 15A15;
 41A45;
 65F20
 EPrint:
 29 pages, 1 figure, 3 tables, submitted to Linear Algebra and its Applications