Singular value decomposition in additive, multiplicative, and logistic forms

摘要

Singular value decomposition (SVD) is widely used in data processing, reduction, and visualization. Applied to a positive matrix, the regular additive SVD by the first several dual vectors can yield irrelevant negative elements of the approximated matrix. We consider a multiplicative SVD modification that corresponds to minimizing the relative errors and produces always positive matrices at any approximation step. Another logistic SVD modification can be used for decomposition of the matrices of proportions, when a regular SVD can yield the elements beyond the zero-one range, while the modified SVD decomposition produces all the elements within the correct range at any step of approximation. Several additional modifications of matrix approximation are also considered.