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 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.