  The original Broad Learning System (BLS) on new added nodes and its existing
efficient implementation both assume the ridge parameter lambda -> 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
lambda -> 0 is no longer assumed, and lambda can be any positive real number.
One of the proposed ridge solutions computes the output weights from the
inverse Cholesky factor, which is updated efficiently 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's method that is a classical tool to compute the
generalized inverse of partitioned matrices. For the proposed efficient ridge
solution based on the inverse Cholesky factor, we also develop another
implementation that is numerically more stable when the ridge parameter lambda
is very small. The proposed ridge solution based on the ridge inverse and the
numerically more stable implementation of the proposed efficient ridge solution
require the same complexity as the original BLS and the existing efficient BLS,
respectively. Moreover, the speedups of the proposed efficient ridge solution
to the original BLS and the existing efficient BLS are 3 and more than 1.67
respectively, when the computational complexities for each update are compared,
and the speedups are 1.99 - 2.52 and 1.31 - 1.58, respectively, when the total
training time is compared by numerical experiments. On the other hand, our
numerical experiments show that both the proposed ridge solutions for BLS
achieve better testing accuracies than the original BLS and the existing
efficient BLS.
