Derivative of L-1 norm of matrix

Question

Assume you want to find the derivative respect to X ($p \times p$) matrix of $$ \frac{\partial}{\partial X} || X - A ||_1 $$

where A is ($p \times p$) matrix.

How can I do it?

Answer 1

Define $$\eqalign{ Y &= (X-A) \cr B &= {\rm abs}(Y) \cr G &= {\rm signum}(Y) \cr B &= Y\odot G \cr }$$ where the functions are applied element-wise.

Then find the differential and gradient of the norm as $$\eqalign{ \phi &= 1:B = 1:Y\odot G = G:Y \cr d\phi &= G:dY = G:dX \cr \frac{\partial\phi}{\partial X} &= G = {\rm signum}\big(X-A\big) \cr\cr }$$ In the above, the symbols {$\,:\,, \odot$} are used to denote the {Frobenius, Hadamard} products, respectively.

Also note that signum has a discontinuity at zero, where its value jumps between $-1$ and $+1$.