Algorithm 1 Low Rank regularization on Class-level Sparse joint representation model (LRCS) and its kernelized version (K-LRCS) for Image Set Matching (ISM)

Input : Learned dictionary D, query set feature matrix Y and balance parameters $\lambda$ and $\gamma$. For the kernelized version, choose proper kernel and its corresponding parameters. Output : Representation coefficient matrix W. Initialization: $W^0=P^0=T^0=\mathrm{O}$, ${\mu }^0={10}^{-6}$, ${\mu }^{max}={10}^8$, ${\rho }_0=1.5$, ${\epsilon }_1={\epsilon }_2={10}^{-6}$. Repeat $(k=0,1,\dots )$ 1:Update W by solving (11), where, for problem (5), ${\nabla }_{W}q\left(W^k\right)=D^{T}D{W}^k-D^{T}Y-T^k+{\mu }^k(W^k-P^k)$ and ${\eta }^k={\mu }^k+{∥D∥}_2^2$ for problem (17), ${\nabla }_{W}q\left(W^k\right)=\frac{1}{2}(K_{D,D}+{K_{D,D}}^T)W^k-{K_{Y,D}}^T-T^k+{\mu }^k(W^k-P^k)$ and ${\eta }^k={\mu }^k+\frac{1}{2}(K_{D,D}+{K_{D,D}}^T)$ 2:update P by solving (14); 3:update T by solving (15); 4:update $\mu$, where ${\mu }^{k+1}=min({\mu }^{max},\rho {\mu }^k)$, where $\rho ={\rho }_0$ if $max\left\{∥W^{k+1}-W^k∥/∥D∥,∥P^{k+1}-P^k∥/∥D∥\right\}\le {\epsilon }_2$; otherwise $\rho =1$; Until $∥W^{k+1}-P^{k+1}∥/∥D∥<{\epsilon }_1$ and $max\left\{∥W^{k+1}-W^k∥/∥D∥,∥P^{k+1}-P^k∥/∥D∥\right\}<{\epsilon }_2$, where ${\epsilon }_2<$