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This paper proposes the recursive and square-root BLS algorithms to improvethe original BLS for new added inputs, which utilize the inverse and inverseCholesky factor of the Hermitian matrix in the ridge inverse, respectively, toupdate the ridge solution. The recursive BLS updates the inverse by the matrixinversion lemma, while the square-root BLS updates the upper-triangular inverseCholesky factor by multiplying it with an upper-triangular intermediate matrix.When the added p training samples are more than the total k nodes in thenetwork, i.e., p>k, the inverse of a sum of matrices is applied to take asmaller matrix inversion or inverse Cholesky factorization. For the distributedBLS with data-parallelism, we introduce the parallel implementation of thesquare-root BLS, which is deduced from the parallel implementation of theinverse Cholesky factorization.  The original BLS based on the generalized inverse with the ridge regressionassumes the ridge parameter lamda->0 in the ridge inverse. When lambda->0 isnot satisfied, the numerical experiments on the MNIST and NORB datasets showthat both the proposed ridge solutions improve the testing accuracy of theoriginal BLS, and the improvement becomes more significant as lambda is bigger.On the other hand, compared to the original BLS, both the proposed BLSalgorithms theoretically require less complexities, and are significantlyfaster in the simulations on the MNIST dataset. The speedups in total trainingtime of the recursive and square-root BLS algorithms over the original BLS are4.41 and 6.92 respectively when p > k, and are 2.80 and 1.59 respectively whenp < k.

Two Efficient Ridge Solutions for the Incremental Broad Learning System on Added Inputs