Say, I have some convex objective function w.r.t an optimization value vector x(e.x linear w.r.t x)
I'd like to minimize the objective plus, make outer product of x with itself to be similar to given positive semidefinite matrix X (i.e $xx^T \sim X$)
To make it, I want a suitable convex loss(penalty) function
Is there any suitable convex loss function?
it's okay if you can make a convex problem by adding some constraint though the loss itself is not always convex
(I checked frobenius norm but the hessian of $ ||X-xx^T||$ gives $-4X+4(x^Tx)I+8xx^T$. Thus it is not convex. If I addition the constraint $X \le (x^Tx)I+2xx^T$, the loss function is convex in the feasible set. However, the constraint is not suit to a convex problem because right hand side is convex function. Constraint for a convex problem should be (convex function <= 0) form )