Why is the $L_{2, 1} $ norm not smooth?

Question

I was reading this paper. I didn't understand why is the $L_{2, 1}$ norm not smooth?

Answer 1

Because it's not differentiable when any row of $W$ is zero. $$\|W\|_{2,1} = \sum_i \left( \sum_j w_{ij}^2 \right)^{1/2}$$ When it is defined, the partial derivative with respect to an individual element of $W$ is $$\frac{\partial}{\partial w_{ij}} \|W\|_{2,1} = w_{ij} \left( \sum_k w_{ik}^2 \right)^{-1/2}$$