In this paper, we propose an efficient algorithm to accelerate the existing Broad Learning System (BLS) algorithm for new added nodes. The existing BLS algorithm computes the output weights from the pseudoinverse with the ridge regression approximation, and updates the pseudoinverse iteratively. As a comparison, the proposed BLS algorithm computes the output weights from the inverse Cholesky factor of the Hermitian matrix in the calculation of the pseudoinverse, and updates the inverse Cholesky factor efficiently. Since the Hermitian matrix in the definition of the pseudoinverse is smaller than the pseudoinverse, the proposed BLS algorithm can reduce the computational complexity, and usually requires less than 2 3 of complexities with respect to the existing BLS algorithm. Our experiments on the Modified National Institute of Standards and Technology (MNIST) dataset show that the speedups in accumulative training time and each additional training time of the proposed BLS over the existing BLS are 24.81%∼ 37.99% and 36.45%∼ 58.96%, respectively, and the speedup in total training time is 37.99%. In our experiments, the proposed BLS and the existing BLS both achieve the same testing accuracy when the tiny differences (≤ 0.05%) caused by the numerical errors are neglected, and the above-mentioned tiny differences and numerical errors become zeroes and ignorable, respectively, when the ridge parameter is not too small. INDEX TERMS Broad learning system (BLS), incremental learning, added nodes, random vector functional-link neural networks (RVFLNN), single layer feedforward neural networks (SLFN), efficient algorithms, partitioned matrix, inverse Cholesky factorization, pseudoinverse.

By combining the broad learning and a convolutional neural network (CNN), a block-diagonal constrained multi-stage convolutional broad learning (MSCBL-BD) method is proposed for hyperspectral image (HSI) classification. Firstly, as the linear sparse feature extracted by the conventional broad learning method cannot fully characterize the complex spatial-spectral features of HSIs, we replace the linear sparse features in the mapped feature (MF) with the features extracted by the CNN to achieve more complex nonlinear mapping. Then, in the multi-layer mapping process of the CNN, information loss occurs to a certain degree. To this end, the multi-stage convolutional features (MSCFs) extracted by the CNN are expanded to obtain the multi-stage broad features (MSBFs). MSCFs and MSBFs are further spliced to obtain multi-stage convolutional broad features (MSCBFs). Additionally, in order to enhance the mutual independence between MSCBFs, a block diagonal constraint is introduced, and MSCBFs are mapped by a block diagonal matrix, so that each feature is represented linearly only by features of the same stage. Finally, the output layer weights of MSCBL-BD and the desired block-diagonal matrix are solved by the alternating direction method of multipliers. Experimental results on three popular HSI datasets demonstrate the superiority of MSCBL-BD.