The maximum residual block Kaczmarz algorithm based on feature selection

Based on the K-means method, an effective block-row partitions algorithm was proposed in ^[1], which involves partitioning the rows of the coefficient matrix $A\in {\mathbb{R}}^{m\times n}$. However, with the increase of the size of the coefficient matrix, the time required for the partitioning process will increase significantly. To address this problem, we considered selecting features from the columns of the matrix $A$ to obtain a low-rank matrix $\tilde{A}\in {\mathbb{R}}^{m\times d}\left(d\ll n\right)$. Lasso is a regression analysis method for feature selection, which is simple and has excellent processing ability for high-dimensional data. In view of this, we first introduced a new criterion for selecting the projection block, and proposed the maximum residual block Kaczmarz algorithm. Then, we put forward the feature selection algorithm based on Lasso, and further presented a maximum residual block Kaczmarz algorithm based on feature selection. We analyzed the convergence of these algorithms and demonstrated their effectiveness through numerical results, while also verifying the performance of the proposed algorithms in image reconstruction.