How to minimize a simple quadratic function of multiple variables ?

[darwid]

Hi everybody,

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 = (y₁,...,y_n) and a vector x = (x₁,...,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 = \sum(y_ix_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

 

[gammamcc]

I wouldn't expect a simple answer. I would liken this to linear regression; cf. http://en.wikipedia.org/wiki/Linear_regression