There is a objective function:

$f(W)$ = $||W||_{2,1}$

For any element $w_{ab}$ in $W$, we apply $F_{w_{ab}}$ to denote the part of $f(W)$ which is only related to $w_{ab}$.

$F'_{w_{ab}}=(DW)_{ab}$

**$F''_{w_{ab}}=(D-D^{3}(W\odot W))_{aa}$,** (the main problem)

where $D_{ii}$=$\frac{1}{||w^i||_2}$, $\odot$ denotes the element-wise multiplication.

About $F_{w_{ab}}$: why the second order derivative of F is equation (28)?

Note that: The norm $\|\cdot\|_{2,1}$ of a matrix $W\in\mathbb{R}^{n\times m}$ is defined as

$$ \Vert W \Vert_{2,1} = \sum_{i=1}^n \Vert w^{i} \Vert_2 = \sum_{i=1}^n \left( \sum_{j=1}^m |w_{ij}|^2 \right)^{1/2} $$ where $w^i$ denotes $i^{th}$ row of $W$, $w_{ij}$ denotes a element of $W$.

**How to solve $F''_{w_{ab}}$? I want to know the detailed calculation process of solving the above formula.**

There some more explicit definition in the following papers (I gave the exact location.)

Some Related Papers:

*Efficient and Robust Feature Selection via Joint $l_{2,1}$-Norms Minimization*

*Graph Regularized Nonnegative Matrix Factorization for Data Representation*
(Look Page 10, APPENDIX A, PROOFS OF THEOREM 1, formulas (26), (27) and (28))

In *Nonnegative matrix factorization by joint locality-constrained and $l_{2,1}$-norm regularization*, (Look Page 7, the Proof of Lemma 1: **Here, how to obtain the result of $F^{''}_{v_{ab}}$?**)

Thank you all for your help.