Rectangular maximum-volume submatrices and their applications

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 maximum-volume submatrices to the case of rectangular maximal-volume 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 low-rank matrices and preconditioning of overdetermined linear systems. The code is available online.