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What is the gradient of $\lVert B-A\circ X\lVert_1$ with respect to $X$. $\circ$ is the hadamard product. $A,B$ are constants

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Let $M=(B-A\circ X)$, then $$\eqalign{ d\,\|M\|_1 &= {\rm sign}(M):dM \cr &= {\rm sign}(M):(-A\circ dX) \cr &= -{\rm sign}(M)\circ A:dX \cr \frac{\partial\,\|M\|_1}{\partial X} &= -A\circ{\rm sign}(M) \cr }$$ in the case of the entrywise (Manhattan) norm. And the sign function is applied entrywise.

If instead you meant the Schatten (Nuclear) norm, then $$\eqalign{ d\,\|M\|_1 &= M(M^TM)^{-1/2}:dM \cr &= M(M^TM)^{-1/2}:(-A\circ dX) \cr &= -(M(M^TM)^{-1/2})\circ A:dX \cr \frac{\partial\,\|M\|_1}{\partial X} &= -A\circ\Big(M(M^TM)^{-1/2}\Big) \cr }$$

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