MATLAB: Minimization problem – matrix frobenius norm

frobenius normoptimization

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

Best Answer

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

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MATLAB: Minimization problem involving matrix norm

I'm assuming the norm in question is the Frobenius norm. When appropriately scaled, the 2D FFT is an orthogonal operator. The problem can then be rewritten

    minimize norm(Bx - FtR), subject to x >= 0

where FtR is the 2D orthonormalized inverse FFT of R.

From that point onward, the NxN shape of the original matrices becomes irrelevant. You can rewrite the problem in terms of basis vectors instead of basis matrices. Specifically, you can write the operator B in explicit matrix form just by reshaping your basis matrices B_i into column vectors and making those the columns of an N^2 x M matrix where M is the number of basis matrices,

    B=[B_1(:),  B_2(:),...B_M(:)];

You would reshape FtR similarly. The minimization problem can then be solved in a straightforward manner either with lsqlin or lsqnonneg.

MATLAB: Solving for scalar in matrix norm minimization

If you have the Optimization Toolbox, you could also use fminimax, although that might be overkill for a simple scalar problem. Recall that the L1-norm of a matrix is its maximum absolute row sum.

Best Answer

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MATLAB: Minimization problem involving matrix norm

MATLAB: Solving for scalar in matrix norm minimization

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