Optimization with respect to matrix

Hi,

I have the following problem to solve: argmin_X || Xu - b ||^2 where u and b are given (vectors), and || . || is l-2 norm. So it looks like ordinary least squares but with optimization that goes over matrix X. Is it possible to do this in Matlab?

[Walter Roberson]

What does armin_X represent? something-something-minimum with respect to X ? Or is X || Xu-b ||^2 indicating X * || X*u - b || ^2 and the armin_ representing something?

[Mateusz]

It should be argmin, not armin.

argmin_X represents minimization with respect to X.

So argmin_X f(X) means minimization of the objective f with respect to X.

[Chaowei Chen]

what is the dimension of u and b? But generally I think you will get infinite many solutions giving you || X*u - b || ^2 =0.

For example, say u and b are both 2D vectors. X is therefore 2 by 2 matrix, having 4 degree of freedom to let you assign values. You can arbitrarily choose x_11 and x_12 such that u_1*x_11+u_2*x_12=b_1 and u_1*x_21+u_2*x_22=b_2. Therefore, the 2-norm is zero.

Answer 1

[Daniel Shub]

The l^2-norm does not care about the shape of your data so you can just reshape your matrix into a vector.

u = u(:);
b = b(:);

[Mateusz]

And the how to reshape my vectors, u and b?

[Daniel Shub]

see my edit.

[Mateusz]

Well, I think it doesn't make sense.

u and b are vectors. X is a matrix.

If X := X(:) and do nothing with u and b (u(:) do nothing with vectors), then I have problems with dimensions (u and b has different dimensionality).