Matrix entropy for an SDP matrix

Dear All,

I would like to compute the entropy of a positive definite matrix M, as defined by the entropy of the vector formed by the (positive) eigenvalues of M.

The corresponding matlab command is pretty simple, namely sum( entr( eig( M ) ) ).
However, I do not manage to adapt this expression to fit the CVX framework.

Would you have any idea on how to proceed?

Best regards and thanks in advance for your help.

I am reasonably certain it cannot be done. Hopefully someone else will respond and prove me wrong. :slight_smile:

Thanks for your comment mcg.
It is quite unfortunate, since this type of spectral functions (provably convex or concave depending on the sign put in front of the entropy) is definitely of interest in many problems.
Let us wait some other responses :smiley:

We actually discussed matrix entropy at some point. The problem is that CVX’s solvers require problems to be representable as SDPs. To our knowledge there is no way to represent matrix entropy using SDP. I am reasonably sure that cannot be done, at least not exactly.

How about iteratively computing a tight lower bound of the entropy (as in EM) and expressing that with CVX? (that should be doable)