To accelerate the existing Broad Learning System (BLS) for new added nodes in [7], we extend the inverse Cholesky factorization in [10] to deduce an efficient inverse Cholesky factorization for a Hermitian matrix partitioned into 2 * 2 blocks, which is utilized to develop the proposed BLS algorithm 1. The proposed BLS algorithm 1 compute the ridge solution (i.e, the output weights) from the inverse Cholesky factor of the Hermitian matrix in the ridge inverse, and update the inverse Cholesky factor efficiently. From the proposed BLS algorithm 1, we deduce the proposed ridge inverse, which can be obtained from the generalized inverse in [7] by just change one matrix in the equation to compute the newly added sub-matrix. We also modify the proposed algorithm 1 into the proposed algorithm 2, which is equivalent to the existing BLS algorithm [7] in terms of numerical computations. The proposed algorithms 1 and 2 can reduce the computational complexity, since usually the Hermitian matrix in the ridge inverse is smaller than the ridge inverse. With respect to the existing BLS algorithm, the proposed algorithms 1 and 2 usually require about 13 and 2 3 of complexities, respectively, while in numerical experiments they achieve the speedups (in each additional training time) of 2.40 - 2.91 and 1.36 - 1.60, respectively. Numerical experiments also show that the proposed algorithm 1 and the standard ridge solution always bear the same testing accuracy, and usually so do the proposed algorithm 2 and the existing BLS algorithm. The existing BLS assumes the ridge parameter lamda->0, since it is based on the generalized inverse with the ridge regression approximation. When the assumption of lamda-> 0 is not satisfied, the standard ridge solution obviously achieves a better testing accuracy than the existing BLS algorithm in numerical experiments.

This brief proposes two BLS algorithms to improve the existing BLS for new added inputs in [7]. The proposed BLS algorithms avoid computing the ridge inverse, by computing the ridge solution (i.e., the output weights) from the inverse or the inverse Cholesky factor of the Hermitian matrix in the ridge inverse. The proposed BLS algorithm 1 updates the inverse of the Hermitian matrix by the matrix inversion lemma [12]. To update the upper-triangular inverse Cholesky factor of the Hermitian matrix, the proposed BLS algorithm 2 multiplies the inverse Cholesky factor with an upper-triangular intermediate matrix, which is computed by a Cholesky factorization or an inverse Cholesky factorization. Assume that the newly added input matrix corresponding to the added inputs is p * k, where p and k are the number of added training samples and the total node number, respectively. When p > k, the inverse of a sum of matrices [11] is utilized to compute the intermediate variables by a smaller matrix inverse in the proposed algorithm 1, or by a smaller inverse Cholesky factorization in the proposed algorithm 2. Usually the Hermitian matrix in the ridge inverse is smaller than the ridge inverse. Thus the proposed algorithms 1 and 2 require less flops (floating-point operations) than the existing BLS algorithm, which is verified by the theoretical flops calculation. In numerical experiments, the speedups for the case of p > k in each additional training time of the proposed BLS algorithms 1 and 2 over the existing algorithm are 1.95 - 5.43 and 2.29 - 6.34, respectively, and the speedups for the case of p < k are 8.83 - 10.21 and 2.28 - 2.58, respectively.

In this brief, we improve the Broad Learning System (BLS) [7] by reducing the computational complexity of the incremental learning for added inputs. We utilize the inverse of a sum of matrices in [8] to improve a step in the pseudoinverse of a row-partitioned matrix. Accordingly we propose two fast algorithms for the cases of q > k and q < k, respectively, where q and k denote the number of additional training samples and the total number of nodes, respectively. Specifically, when q > k, the proposed algorithm computes only a k * k matrix inverse, instead of a q * q matrix inverse in the existing algorithm. Accordingly it can reduce the complexity dramatically. Our simulations, which follow those for Table V in [7], show that the proposed algorithm and the existing algorithm achieve the same testing accuracy, while the speedups in BLS training time of the proposed algorithm over the existing algorithm are 1.24 - 1.30.

The decremented learning algorithms are required in machine learning, to prune redundant nodes and remove obsolete inline training samples. In this paper, an efficient decremented learning algorithm to prune redundant nodes is deduced from the incremental learning algorithm 1 proposed in [9] for added nodes, and two decremented learning algorithms to remove training samples are deduced from the two incremental learning algorithms proposed in [10] for added inputs. The proposed decremented learning algorithm for reduced nodes utilizes the inverse Cholesterol factor of the Herminia matrix in the ridge inverse, to update the output weights recursively, as the incremental learning algorithm 1 for added nodes in [9], while that inverse Cholesterol factor is updated with an unitary transformation. The proposed decremented learning algorithm 1 for reduced inputs updates the output weights recursively with the inverse of the Herminia matrix in the ridge inverse, and updates that inverse recursively, as the incremental learning algorithm 1 for added inputs in [10].

This paper considers the sparse generalized eigenvalue problem (SGEP), which aims to find the leading eigenvector with at most $k$ nonzero entries. SGEP naturally arises in many applications in machine learning, statistics, and scientific computing, for example, the sparse principal component analysis (SPCA), the sparse discriminant analysis (SDA), and the sparse canonical correlation analysis (SCCA). In this paper, we focus on the development of a three-stage algorithm named {\em inverse-free truncated Rayleigh-Ritz method} ({\em IFTRR}) to efficiently solve SGEP. In each iteration of IFTRR, only a small number of matrix-vector products is required. This makes IFTRR well-suited for large scale problems. Particularly, a new truncation strategy is proposed, which is able to find the support set of the leading eigenvector effectively. Theoretical results are developed to explain why IFTRR works well. Numerical simulations demonstrate the merits of IFTRR.