A New Improvement Of Davidson 's Method By Random Low Rank Decomposition

Principal component analysis and its kernal promotion have a wide range of applications in many areas, however, when the indicators or the number of samples of the data is enormous, the size of the covariance matrix or kernal matrix will be very large, leading to low computational efficiency if we use eigenvalue decomposition directly. Taking into account that the principal component analysis often requires only k dominant components, in order to reduce the computation time of the algorithm as much as possible on the basis of the accuracy of results, we analyzed the Davidson method and introduced a randomized algorithm for constructing low-rank matrix decomposition, and then combined the randomized algorithm with the Davidson method to construct a new improved algorithm to get the approximate solution of the k largest eigenvalues and their corresponding eigenvectors of the original matrix, in order to compensate the problem of the Davidson method that the size of the iterative matrix may grow rapidly during the iterative process which will lead to a decline in the efficiency of the algorithm. The theoretical analysis and numerical calculation results showed that the improved algorithm had a good performance whether in the reliability of the results of the analysis or the computational efficiency. Especially when we are using the principal component analysis in the face of high-dimensional data, the improved algorithm will show a fairly good value.