Consider the following function $f(\bf x)$: $f(\bf x) = \left\|(\bf A \bf x) \circ {(\bf A \bf x)}^* - \bf c\right\|_2^2$, where
- $\bf A \in {\mathbb C}^{N\times M}$ (complex-valued matrix)
- $\bf x \in {\mathbb C}^{M \times1}$ (complex-valued vector)
- $\bf c \in {\mathbb R}^{N \times1}$ (real-valued vector)
- $\left\| \right\|_2$ denotes the $l_2$ norm
- $\circ$ denotes the Hadamard product
- $^*$ denotes the conjugate operation.
The aim is to compute the derivative of $f(\bf x)$ with respect to $\bf x$.
This might involve some notion and techniques of complex-valued matrix differentiation, which I am not familiar. Can anyone point out some references on solving this problem?
Thanks.