Solve for diagonal matrix D by minimizing the operator norm in Matlab

Say that the size of matrix is ​​576x576 and size of F is1296x576.

Which matlab function can I use to solve this problem?

[Matt J]

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Answer 1

[Bruno Luong]

fminunc, fmincon and family.

[christina]

How exactly? Could you please give me an example?

[Torsten]

A = ...;
F = ...;
D0 = ones(n,1)
D = fminunc(@(x)fun(x,A,F),D0)

function obj = fun(D,A,F)
obj = norm(A-F'*diag(D)*F,2)
% obj = norm(A-F'*diag(D)*F,inf)
end

[christina]

Could you please clarify why do your write diag(D) instead of just D while defining the objective function?

[Bruno Luong]

fminunc (and all optimization function) accept vector as input, DIAG transform to matrix.

[christina]

Thanks for the response. So you mean, fminunc vectorizes the matrix before solving? If it vectorizes the matrix, then I am not sure if minimizing the 2-norm of a vector is same as minimizing the 2-norm of a matrix. In that case, do you have any idea how do I solve this problem? t

[Torsten]

You misunderstood. The unknowns for all optimization codes have to be passed to the optimizer (in this case fminunc) as a vector. I chose the main diagonal of D as this vector d. In order to perform the matrix multiplication F'*D*F, the vector d has to be transformed to a diagonal matrix D. This is done via the command "diag(d)".

Maybe the procedure becomes clearer if I alter the code to

A = ...;
F = ...;
d0 = ones(n,1)
d = fminunc(@(x)fun(x,A,F),d0)

function obj = fun(d,A,F)
D = diag(d);
obj = norm(A-F'*D*F,2)
% obj = norm(A-F'*D*F,inf)
end

[Bruno Luong]

No. The standard math definition of VECTORIZE the matrix is

v = M(:)
M = reshape(v,[m,m])

Here

v = diag(M)
M = diag(v)

This is NOT vectorize.

FMINCON don't do anything beside minimize an objective function that user defines and provides. In the case Torsen's code

norm(A-F'*diag(D)*F,2)

is matrix 2-norm (maximum singular value).

Torsten's code is correct and do not need any modification.

You however needs to read careful the doc of fminunc, diag, norm.

[christina]

Thank you so much. It is clear now.