Fast optimization of common basis for matrix set through Common Singular Value Decomposition

Abstract

SVD (singular value decomposition) is one of the basic tools of machine learning, allowing to optimize basis for a given matrix. However, sometimes we have a set of matrices _k\_k instead, and would like to optimize a single common basis for them: find orthogonal matrices U, V, such that ^T A_k V\ set of matrices is somehow simpler. For example DCT-II is orthonormal basis of functions commonly used in image/video compression - as discussed here, this kind of basis can be quickly automatically optimized for a given dataset. While also discussed gradient descent optimization might be computationally costly, there is proposed CSVD (common SVD): fast general approach based on SVD. Specifically, we choose U as built of eigenvectors of _i (w_k)^q (A_k A_k^T)^p and V of _k (w_k)^q (A_k^T A_k)^p, where w_k are their weights, p,q>0 are some chosen powers e.g. 1/2, optionally with normalization e.g. A A - rc^T where r_i=_j A_ij, c_j =_i A_ij.

Tasks

Reproductions