Input: Data set $X\in R^{D\times n}$ ; the reduced dimension d, parameter p, $\sigma$ . Output: Coefficient matrix $\alpha \in R^{n\times d}$ .

1: Construct the kernel matrix $K$ ;

2: Decompose $K$ by using eigenvectors decomposition;

3: Initialize $\alpha$ by using KDA [42] ;

4: Repeat step 5-step 9;

5: Update $W_{\phi k}$ base on (13) ;

6: Update $d_{ij}^{\phi }={\Vert W_{\phi }^{\text{T}}\left(\phi \left(x_i\right)-\phi \left(x_j\right)\right)\Vert }_2^{p-2}$ ;

7: Update $\beta$ by using the eigenvalue decomposition of (18);

8: Update $\alpha$ by $\alpha =U{\Gamma }^{-1}\beta$ ;

9: $t=t+1$ ;

10: Until convergence;

11: Return coefficient matrix $\alpha$ .