# How to minimize a simple quadratic function of multiple variables ?

1. Mar 30, 2009

### 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,...,yn) and a vector x = (x1,...,xm) such that their outer product approximates a given matrix A = (ai,j), i=1..n, j=1..m

What I want to minimize is thus:
s = $$\sum$$(yixj-ai,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

2. Apr 5, 2009