Algorithm 1: The RM-GNMF-based Algorithm

Input: The dataset $X=[x_1,x_2,\cdots ,x_n]\in R^{m\times n}$,  a predefined number of clusters k,  parameters $\mu$, $\lambda$,  the nearest neighbor graph parameter p,  maximum iteration number $t_{max}$.  Initialization:  $Z=\mathsf{\Lambda }=0$, $W_0\in R^{m\times k}$, $H_0\in R^{k\times n}$.  Repeat  Fix other parameters, and then update Z by formula (17);  Fix other parameters, and then update W by:  $W=(X-Z+\frac{\mathsf{\Lambda }}{\mu })H$;  Update H by $H=U{V}^T$, where U and V are  the left and right singular values of the SVD decomposition;  Fix other parameters, and then update $\mathsf{\Lambda }$ and $\mu$ by formulas (26)(27);  t = t + 1;  Until $t\ge t_{max}$.  Output: matrix $W\in R^{m\times k}$ , matrix $H\in R^{k\times n}$.