To accelerate the existing Broad Learning System (BLS) for new added nodes in , we extend the inverse Cholesky factorization in  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  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  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 . 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 . 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  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.
Inverse kinematics solves the problem of how to control robot arm joints to achieve desired end effector positions, which is critical to any robot arm design and implementations of control algorithms. It is a common misunderstanding that closed-form inverse kinematics analysis is solved. Popular software and algorithms, such as gradient descent or any multi-variant equations solving algorithm, claims solving inverse kinematics but only on the numerical level. While the numerical inverse kinematics solutions are relatively straightforward to obtain, these methods often fail, due to dependency on specific numerical values, even when the inverse kinematics solutions exist. Therefore, closed-form inverse kinematics analysis is superior, but there is no generalized automated algorithm.
Gaussian graphical models are of great interest in statistical learning. In this paper, we propose a first-order method based on an alternating linearization technique that exploits the problem's special structure; in particular, the subproblems solved in each iteration have closed-form solutions. Numerical experiments on both synthetic and real data from gene association networks show that a practical version of this algorithm outperforms other competitive algorithms. Papers published at the Neural Information Processing Systems Conference.
Sparse graphical modelling/inverse covariance selection is an important problem in machine learning and has seen significant advances in recent years. A major focus has been on methods which perform model selection in high dimensions. It is not however clear which of these methods are optimal in any well-defined sense. A major gap in this regard pertains to the rate of convergence of proposed optimization methods. The proximal gradient method considered in this paper is shown to converge at a linear rate, a result which is the first of its kind for numerically solving the sparse inverse covariance estimation problem.