Derivative of $tr(A(C \circ X)BB'(C' \circ X')A')$

Question

Can we differentiate this function: $tr(A(C \circ X)BB'(C' \circ X')A')$ w.r.t $X$?

Also, $tr(A(C \circ X)Y)$ w.r.t $X$.

Answer 1

The Frobenius product is a convenient notation for the trace $\,A:B = {\rm Tr}(A^TB)$

Define the matrices $$\eqalign{ Y &= C\circ X &\implies dY &= C\circ dX \cr W &= AYB &\implies dW &= A\,dY\,B \cr }$$ Write the function in terms of these new variables. Then calculate its differential and gradient. $$\eqalign{ \phi &= W:W \cr d\phi &= W:dW + dW:W \cr &= 2W:dW \cr &= 2W:(A\,dY\,B) \cr &= 2(A^TWB^T):dY \cr &= 2(A^TWB^T):(C\circ dX) \cr &= 2\big(C\circ (A^TWB^T)\big):dX \cr &= 2\Big(C\circ\big(A^TA(C\circ X)BB^T\big)\Big):dX \cr \frac{\partial\phi}{\partial X} &= 2C\circ\big(A^TA(C\circ X)BB^T\big) \cr }$$