I have previously asked this question but did not get a response. Maybe it was because tag was wrong. I deleted the previous one and asking this again. I have this cost function: $$f(X)=||AX||_{2,1}$$ I want to derive $$\nabla_X f(X)$$ I have done this derivation: $$[AX]_{ij}=\sum_{k}A_{ik}X_{kj}$$ $$||AX||_{2,1}=\sum_{i}(\sum_{j}(\sum_{k}A_{ik}X_{kj})^2)^{1/2}$$ $$\frac{\partial}{\partial X_{ij}}||AX||_{2,1}=\sum_{i}(\frac{1}{(\sum_{m}(\sum_{l}A_{il}X_{lm})^2)^{1/2}})\sum_{j}(\sum_{k}A_{ik}X_{kj})\frac{\partial}{\partial X_{ij}}\sum_{k}A_{ik}X_{kj}$$ $$[\nabla_{X}||AX||_{2,1}]_{ij}=\frac{(\sum_kA_{i,k}X_{kj})A_{ii}}{(\sum_m(\sum_lA_{il}X_{lm})^2)^{1/2}}$$
Does it seem ok to you? Have a nice day.