Algorithm 1 Dictionary Learning (K-SVD)

Input: Example $Y={\left\{y_i\right\}}_{i=1}^N$, initialize dictionary D, sparse matrix X, Output: Dictionary, sparse matrix. 1: Initialize: Randomly take K column vectors from the original sample $Y\in R^{m\times n}$ or take the first K column vectors $d_1,d_2,\cdots ,d_K$ of its left singular matrix as the atoms of the initial dictionary, and the dictionary $D_0\in R^{m\times K}$, $j=0$, maximum iterations J, tolerance value ${\epsilon }_0$. 2: Sparse coding: Using the dictionary $D_j$, $X_j\in R^{K\times n}$ is obtained by $$\underset{D,X}{min}{∥Y-DX∥}_F^{2}s.t.\forall i,{∥x_i∥}_0\le {\epsilon }_0.$$ (2) 3: Dictionary update: Update dictionary $D_j$ column by column, column $d_k\in \left\{d_1,d_2,\cdots ,d_K\right\}$. When updating $d_k$, calculate the error matrix $E_k$, $E_k=Y-{\displaystyle \sum _{j\ne k}}{d}_j{x}_T^j$; Take out the set of indices where the k-th row vector $x_T^k$ of the sparse matrix is not 0, ${\omega }_k=\left\{i|1<i<n,x_T^k\left(i\right)\ne 0\right\}$, $x_T^{{}^{\prime }k}=\left\{x_T^k\left(i\right)|1<i<n,x_T^k\left(i\right)\ne 0\right\}$. Elect the column corresponding to ${\omega }_k\ne 0$ from $E_k$, and obtain $E_k^{{}^{\prime }}$. Perform singular value decomposition of $E_k^{{}^{\prime }}$, $E_k^{{}^{\prime }}=U\Sigma VT$, take the first column of U to update the k-th column of the dictionary, that is, $d_k=U(·,1)$; Let $x_T^{{}^{\prime }k}=\Sigma (1,1)V{(·,1)}^T$, after obtaining $x_T^{{}^{\prime }k}$, update accordingly it to the original $x_T^k$. Set $j=j+1$. 4: repeat 5:     The sparse coding and dictionary update steps; 6: until the specified number of iteration steps J is reached, or converge to the specified error ${\epsilon }_0$.