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The inverse-free extreme learning machine (ELM) algorithm proposed in [4] wasbased on an inverse-free algorithm to compute the regularized pseudo-inverse,which was deduced from an inverse-free recursive algorithm to update theinverse of a Hermitian matrix. Before that recursive algorithm was applied in[4], its improved version had been utilized in previous literatures [9], [10].Accordingly from the improved recursive algorithm [9], [10], we deduce a moreefficient inverse-free algorithm  to update the regularized pseudo-inverse, from which we develop the proposedinverse-free ELM algorithm 1. Moreover, the proposed ELM algorithm 2 furtherreduces the computational complexity, which computes the output weightsdirectly from the updated inverse, and avoids computing the regularizedpseudoinverse. Lastly, instead of updating the inverse, the proposed ELMalgorithm 3 updates the LDLT factor of the inverse by the inverse LDLTfactorization [11], to avoid numerical instabilities after a very large numberof iterations [12]. With respect to the existing ELM algorithm, the proposedELM algorithms 1, 2 and 3 are expected to require only (8+3)/M , (8+1)/M and(8+1)/M of complexities, respectively, where M is the output node number. Inthe numerical experiments, the standard ELM, the existing inverse-free ELMalgorithm and the proposed ELM algorithms 1, 2 and 3 achieve the sameperformance in regression and classification, while all the 3 proposedalgorithms significantly accelerate the existing inverse-free ELM algorithm

Efficient Inverse-Free Algorithms for Extreme Learning Machine Based on the Recursive Matrix Inverse and the Inverse LDL' Factorization