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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.

Mike
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  • I would recommend the book "Complex-Valued Matrix Derivatives" by Are Hjorungnes. He's also published some PDF articles with roughly the same title, which you can Google. – greg Sep 17 '18 at 04:19
  • Thanks. I will read the material you pointed out. – Mike Sep 17 '18 at 12:22
  • $(Ax)\circ (Ax)^$ is not defined except if $$ is the conjugate operation; in this case it's a bad notation. –  Sep 17 '18 at 18:26
  • On the other hand, if $x$ varies in $\mathbb{C}$, then the function $x\rightarrow |x|^2$ has no derivative wrt. $x$. –  Sep 17 '18 at 18:33
  • In terms of Wirtinger derivatives, you can write $$\eqalign{ \frac{\partial f}{\partial x} &= 2A^T(b^\circ b\circ b^-b^\circ c) \cr \frac{\partial f}{\partial x^} &= 2A^H(b\circ b^*\circ b-b\circ c) \cr }$$ where $,b=Ax$ – greg Sep 17 '18 at 22:08

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