If $f(x) = \|AX\|_1$, where $\|.\|$ denotes the entrywise $\ell_1$-norm, what is the subgradient of $f(x)$?
Is there an expression similar to $A'A \ \partial \|X\|_1$? For example, $A = [1 \ 1; 1 \ 0; 0 \ 1]$.
2026-03-25 07:41:16.1774424476
Subgradient of $\|AX\|_1$
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There is a chain rule for subgradients. I guess you are on $\mathbb{R}^n$ and there it holds, for example, for $f(x) = g(Ax)$, that $$ \partial f(x) = A^T\partial g(Ax), $$ see, e.g. Theorem 9.3 here.