Least Square Estimator for Matrices: Bill's Problem

[bhobba]

I recently came across the following interesting problem.

Suppose A = BC where A,B, and C are matrices. We know a ton of A's and their corresponding C's. We want the least square estimator of B.

When A and C are vectors the solution is well known.

But what is the solution when they are matrices?

Thanks
Bill

 

[Stephen Tashi]

How do you intend to define the "error" between the observed and predicted values? Until that is defined, "least squares" doesn't describe a specific criteria.

 

[bhobba]

The matrix L2 norm ie the B that minimises ∑ ||Ai - BCi||^2 where Ai and Ci are the known A C matrix pairs. By matrix L2 norm I mean the generalisation of the usual vector norm ie the square root of the sum of the squares of the matrix elements.

It grew out of the following paper:
http://www.cv-foundation.org/openac...t_Direct_Super-Resolution_2013_ICCV_paper.pdf

See equation 2.

Thanks
Bill

 

[WWGD]

Isn't there an equivalent of a perp projection operator in your space of matrices ?If this space is a Hilbert space, then, AFAIK, the general solution to this problem in a Hilbert space is the ortho. projection of B onto the subspace spanned by A,C.

 

[bhobba]

Isn't there an equivalent of a perp projection operator in your space of matrices ?If this space is a Hilbert space, then, AFAIK, the general solution to this problem in a Hilbert space is the ortho. projection of B onto the subspace spanned by A,C.

Yes there is - its the trace. I will think about that one.

Thanks
Bill

 

[bhobba]

Hi Guys

Thanks for all the help.

Finally nutted it out. As usual I was on the wrong track. It's simply a matter by blocking the problem and reducing it to a number of ordinary least squares problems. Break B into rows Bj so you get the usual least squares problems ||Aji - BjCi||^2. The minimum is the minimum of each of these separate problems.

Thanks
Bill