Efficient and Stable Algorithms to Extend Greville's Method to Partitioned Matrices Based on Inverse Cholesky Factorization

Greville's method has been utilized in (Broad Learn-ing System) BLS to propose an effective and efficient incremental learning system without retraining the whole network from the beginning. For a column-partitioned matrix where the second part consists of p columns, Greville's method requires p iterations to compute the pseudoinverse of the whole matrix from the pseudoinverse of the first part. The incremental algorithms in BLS extend Greville's method to compute the pseudoinverse of the whole matrix from the pseudoinverse of the first part by just 1 iteration, which have neglected some possible cases, and need further improvements in efficiency and numerical stability. In this paper, we propose an efficient and numerical stable algorithm from Greville's method, to compute the pseudoinverse of the whole matrix from the pseudoinverse of the first part by just 1 iteration, where all possible cases are considered, and the recently proposed inverse Cholesky factorization can be applied to further reduce the computational complexity. Finally, we give the whole algorithm for column-partitioned matrices in BLS. On the other hand, we also give the proposed algorithm for row-partitioned matrices in BLS

Recursive LMMSE-Based Iterative Soft Interference Cancellation for MIMO Systems to Save Computations and Memories

Firstly, a reordered description is given for the linear minimum mean square error (LMMSE)-based iterative soft interference cancellation (ISIC) detection process for Mutipleinput multiple-output (MIMO) wireless communication systems, which is based on the equivalent channel matrix. Then the above reordered description is applied to compare the detection process for LMMSE-ISIC with that for the hard decision (HD)-based ordered successive interference cancellation (OSIC) scheme, to draw the conclusion that the former is the extension of the latter. Finally, the recursive scheme for HD-OSIC with reduced complexity and memory saving is extended to propose the recursive scheme for LMMSE-ISIC, where the required computations and memories are reduced by computing the filtering bias and the estimate from the Hermitian inverse matrix and the symbol estimate vector, and updating the Hermitian inverse matrix and the symbol estimate vector efficiently. Assume N transmitters and M (no less than N) receivers in the MIMO system. Compared to the existing low-complexity LMMSE-ISIC scheme, the proposed recursive LMMSE-ISIC scheme requires no more than 1/6 computations and no more than 1/5 memory units

Two Ridge Solutions for the Incremental Broad Learning System on Added Nodes

The original Broad Learning System (BLS) on new added nodes and its existing efficient implementation both assume the ridge parameter is near 0 in the ridge inverse to approximate the generalized inverse, and compute the generalized inverse solution for the output weights. In this paper, we propose two ridge solutions for the output weights in the BLS on added nodes, where the ridge parameter can be any positive real number. One of the proposed ridge solutions computes the output weights from the inverse Cholesky factor, which is updated by extending the existing inverse Cholesky factorization. The other proposed ridge solution computes the output weights from the ridge inverse, and updates the ridge inverse by extending the Greville method that can only computes the generalized inverse of a partitioned matrix. The proposed BLS algorithm based on the ridge inverse requires the same complexity as the original BLS algorithm, while the proposed BLS algorithm based on the inverse Cholesky factor requires less complexity and training time than the original BLS and the existing efficient BLS. Both the proposed ridge solutions for BLS achieve the same testing accuracy as the standard ridge solution in the numerical experiments. The difference between the testing accuracy of the proposed ridge solutions and that of the existing generalized inverse solutions is negligible when the ridge parameter is very small, and becomes too big to be ignored when the ridge parameter is not very small. When the ridge parameter is not near 0, usually the proposed two ridge solutions for BLS achieve better testing accuracy than the existing generalized inverse solutions for BLS, and then the former are more preferred than the latter

Improved Recursive Algorithms for V-BLAST to Reduce the Complexity and Save Memories

Improvements I-IV were proposed to reduce the computational complexity of the original recursive algorithm for vertical Bell Laboratories layered space-time architecture (VBLAST). The existing recursive algorithm with speed advantage and that with memory saving incorporate Improvements I-IV and only Improvements III-IV into the original algorithm, respectively. To the best of our knowledge, the algorithm with speed advantage and that with memory saving require the lowest complexity and the least memories, respectively, among the existing recursive V-BLAST algorithms. We propose Improvements V and VI to replace Improvements I and II, respectively. Instead of the lemma for inversion of partitioned matrix applied in Improvement I, Improvement V uses another lemma to speed up the matrix inversion step by the factor of 1.67. Then the formulas adopted in our Improvement V are applied to deduce Improvement VI, which includes the improved interference cancellation scheme with memory saving. In the existing algorithm with speed advantage, the proposed algorithm I with speed advantage replaces Improvement I with Improvement V, while the proposed algorithm II with both speed advantage and memory saving replaces Improvements I and II with Improvements V and VI, respectively. Both proposed algorithms speed up the existing algorithm with speed advantage by the factor of 1.3, while the proposed algorithm II achieves the speedup of 1.86 and saves about half memories, compared to the existing algorithm with memory saving