On Feb 24, 2016, at 2:13 PM, Dan Asimov <dasimov@earthlink.net> wrote:
It might be worth mentioning that given any way to do this factoring, say
M = A x B
, then for any kxk matrix G having an inverse H we also have
M = (AxG) x (HxB)
This runs into problems when you want to keep the factors non-negative, which was Mike’s question. Fun puzzle: what group of matrices (for insertion) preserves non-negativity?
Also, which ways to find an approximate factorization are best will probably depend on what kind of measure of error is used.
If the error used is what is perhaps the most obvious one, namely
Err(M; A, B) = L_2(M - AxB)
where L_2(N) is defined as the sum of the squared entries of N (or sum of squared absolute values if over the complexes), then probably just a garden-variety optimization algorithm will work well.
(E.g., a conjugate-gradient method from "Practical Optimization" by Murray, Gill, and Wright — a book I highly recommend.)
—Dan