Numerical strategies for recursive least squares solutions to the matrix equation AX = B

The recursive solution to the Procrustes problem -with or without constraints- is thoroughly investigated. Given known matrices A and B, the proposed solution minimizes the square of the Frobenius norm of the difference AX−B when rows or columns are added to A and B. The proposed method is based on efficient strategies which reduce the computational cost by utilizing previous computations when new data are acquired. This is particularly useful in the iterative solution of an unbalanced orthogonal Procrustes problem. The results show that the computational efficiency of the proposed recursive algorithms is more significant when the dimensions of the matrices are large. This demonstrates the usefulness of the proposed algorithms in the presence of high-dimensional data sets. The practicality of the new method is demonstrated through an application in machine learning, namely feature extraction for image processing.