I haven’t looked very closely at what you’re trying to do,. But take a look at Perspective of log det function, and CVX formulations of related log det problems using Quantum Relative Entropy from CVXQUAD
Can you do what you want in a manner along those lines?
However, are you trying to raise non-square matrices to powers? I don’t see how that works unless the matrices are squares, i.e., U = K. Or are you actually trying to do element-wise powers, which is a very different thing, for which matrix functions are not applicable?
In any event, if you are actually looking at matrix (not element-wise) powers, look at the functions (and sets) listed at https://github.com/hfawzi/cvxquad ., which are available in CVX by installing CVXQUAD (you don’t need to install its exponential.m replacement to use the matrix functions). For instance, trace_mpower(X,t,C) = trace(C*X^t),
concave in X for t in [0,1]
convex in X for t in [-1,0] or [1,2]
(C fixed positive semidefinite matrix)`