Algorithm 1: Weighted Kernel Sparse Representation Classifier

1. Hierarchical Features Extraction based on CENTRIST

2. WKSR: Initialize the weight in each block as 1: ${\omega }_i=1$ While not converge, do (a) Compute weighted kernel sparse representation ${\widehat{\alpha }}_i=\arg \underset{\alpha }{\min }\left\{{\displaystyle \sum _{i=1}^B{\omega }_{i}k(y_i,y_i)}+{{\alpha }_i}^T{\displaystyle \sum _{i=1}^B{\omega }_i{k}_{X_i{X}_i}}{\alpha }_i-2{{\alpha }_i}^T{\displaystyle \sum _{i=1}^B{\omega }_i{k}_{X_i{y}_i}}+\lambda {\Vert {\alpha }_i\Vert }_1\right\}$ (b) Compute the reconstruction residual $e_i={\sqrt{\Vert \phi (y_i)-\phi (X_i){\alpha }_i\Vert }}_2^2=k(y_i,y_i)+{{\widehat{\alpha }}_j}^T{k}_{X_i{X}_i}{\widehat{\alpha }}_j-2{{\widehat{\alpha }}_j}^T{k}_{X_i{y}_i}$ (c) Compute the weight value $\omega (e_i)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{{e_i}^2}{2{\sigma }^2}\right)$ (d) Checking convergence condition ${\displaystyle \sum _{i=1}^B({{\omega }_i}^{(t)}-}{{\omega }_i}^{(t-1)}{)}^2/{\displaystyle \sum _{i=1}^B{({{\omega }_i}^{(t-1)})}^2<\tau }$ where $\tau$ is a small positive scalar and ${{\omega }_i}^{(t)}$ is the weight value of ith block in the iteration t.

3. Do classification $\text{identity}=\arg \underset{j}{\min }\left\{{\displaystyle \sum _{i=1}^B{\omega }_{i}k(y_i,y_i)}+{{\widehat{\alpha }}_j}^T{\displaystyle \sum _{i=1}^B{\omega }_i{k}_{X_{i,j}{X}_{i,j}}}{\widehat{\alpha }}_j-2{{\widehat{\alpha }}_j}^T{\displaystyle \sum _{i=1}^B{\omega }_i{k}_{X_{i,j}{y}_i}}\right\},j=0,1$ where $X_{i,j}$ the sub-matrix of $X_i$ associated with the jth class, ${\widehat{\alpha }}_j$ being the representation coefficient vector associated with the jth class.