Approximation?

epostma — Sat, 02 Oct 2010 03:48:15 Z

What a short bike ride can't do for you (from work to home). Here is at least an approximation algorithm, valid for general kth order tensors v.

Let u be the best rank-one approximation to v. If v = u, stop, otherwise approximate v - u recursively.

Now the problem has been reduced to finding the best rank one approximation to a general kth order tensor v. I think we can do this as follows:

Add the entries along one dimension of the tensor, resulting in a (k - 1)th order tensor w. Take the best rank-one tensor approximating w. Find a vector t so that t tensor w is the best possible approximation to v, by considering the components of t individually.

Again a reduction, this time to the same problem for k - 1 (which we solve by induction with SVD as the base case) and to finding a scalar t_i such that t_i * w approximates v_i best. This sounds like a problem that is manageable; I think it's probably convex.

However, this is certainly only an approximation algorithm: for the tensor

tensor minus tensor

every resulting approximation is the zero matrix. I guess I will implement this and see how well it performs for images...

Erik Postma
Maplesoft.

Wikipedia

epostma — Sat, 02 Oct 2010 04:15:15 Z

I can't believe I didn't check wikipedia: http://en.wikipedia.org/wiki/Higher-order_singular_value_decomposition explains that there are two notions of SVD for higher-order tensors, and the one I need is called candecomp-PARAFAC. I guess I'll be implementing that some time next week...

Erik.

MaplePrimes - answers and comments on Question, Decompose a third order tensor into pure tensors?

Approximation?

Wikipedia