I'm trying to minimize a function with multiple variables. My goal is to approximate on the L2 norm a matrix by the outer product of 2 vectors (or is it called tensor product ?).

So I have to determine a vector y = (y

_{1},...,y

_{n}) and a vector x = (x

_{1},...,x

_{m}) such that their outer product approximates a given matrix A = (a

_{i,j}), i=1..n, j=1..m

What I want to minimize is thus:

s = [tex]\sum[/tex](y

_{i}x

_{j}-a

_{i,j})

^{2}

Obviously I can solve this using a gradient descent and it works.

But what I'm looking for is an analytical solution. The formulation looks simple so I expect there must be some analytical way of solving this, it's just that I don't really know how to approach this problem due to the many variables.

--

Darwid