The greedy block Kaczmarz (GBK) method has been successfully applied in areas such as data mining, image reconstruction, and large-scale image restoration. However, the computation of pseudo-inverses in each iterative step of the GBK method not only complicates the computation and slows down the convergence rate, but it is also ill-suited for distributed implementation. The leverage score sampling free pseudo-inverse GBK algorithm proposed in this paper demonstrated significant potential in the field of image reconstruction. By ingeniously transforming the problem framework, the algorithm not only enhanced the efficiency of processing systems of linear equations with multiple solution vectors but also optimized specifically for applications in image reconstruction. A methodology that combined theoretical and experimental approaches has validated the robustness and practicality of the algorithm, providing valuable insights for technical advancements in related disciplines.

Through time series signals collected by wearable sensors, human activity recognition (HAR) needs to be carried out based on the given categories of training samples. However, there always exist new classes of data in real environment. These days how to effectively distinguish these new categories of data from the given classes become an important issue for HAR. Class-incremental learning aims to update the existing model using new knowledge when the target data is increasing. Also, the open set based recognition algorithm can provide the rejection option for the classifier to identify the target class that the model didn’t learn before. In this paper, an open set based class-incremental learning HAR framework is designed, which can continuously identify and learn new unknown classes. The framework combines extreme value model (EVM) with incremental learning to learn and recognizes new data. Here, PCA dimensionality reduction for features is applied to calculates the cosine, Euclidean and Manhattan distances between features, respectively. The simulation result reveals that the proposed model performs well on the UCI and PAMAP2 datasets compared with the existing open set based schemes. Higher accuracy can be achieved through PCA reduction with cosine distance calculation. Also, in the class incremental learning experiment, the proposed model can maintain high accuracy while new classes can be effectively identified.