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.