Minimization problem - matrix frobenius norm

Hey there, i am working on homework and i need to implement an algorithm from a paper. in the final stage, i have to smooth a matrix which represents movment of camera, (9xnumber of frames). this is done by fitting a 3 dim polynomial model, so if E is the camera matrix, i have to solve:

    min ||C*E-C*K*B||

where K is 9*dim coeff matrix, and B is just the variable meshgridded: [1, 1^2, 1^3; 2 2^2 2^3;3 3^2 3^3] etc.

the norm is Frobenius norm. how can i solve this in matlab? i tried lsqnolin, it's working, but taking LONG time, and easily reaches memory limit with more than 400 frames.

Thanks

Answer 1

[Matt J]

So, we're solving for K? Since there are no constraints, it is a purely linear least squares problem and an analytical solution is available,

K=C\(C*E)/B

If the matrix C is square, nonsingular, this of course reduces to

   K=E/B

[Matt J]

Comment by aviel nah

Hey, thanks, it seems you are correct, i was quite sure i cannot use least squares since i am trying to minimize a matrix norm, which is differend than the 2d euclidian norm on 1d vectors.

but how would you add masking to this equation? i mean, i actually forgot to write one matrix in the eq. which is a bit-wise masking matrix, so i am trying to minimize: |W.*(C*E-B*K*E)|. W is size of C*E. how would you add that into the ls solution?

[Matt J]

i was quite sure i cannot use least squares since i am trying to minimize a matrix norm, which is differend than the 2d euclidian norm on 1d vectors.

The Frobenius norm is not an induced matrix norm, so it is actually very much the same. The problem can be rewritten in 1D vector form as follows

   y=reshape(C*E,[],1);

   A=kron(B.',C);

    min ||A*k-y||

where k=K(:).

[aviel nah]

and can you also add the extra Matrix W? which is a binary masking matrix: min |W.*(C*E-C*K*B)|||?