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Linear discriminant analysis (LDA) is a well-known technique for linear classification, feature extraction, and dimension reduction. To improve the accuracy of LDA under the high dimension low sample size (HDLSS) settings, shrunken estimators, such as Graphical Lasso, can be used to strike a balance between biases and variances. Although the estimator with induced sparsity obtains a faster convergence rate, however, the introduced bias may also degrade the performance. In this paper, we theoretically analyze how the sparsity and the convergence rate of the precision matrix (also known as inverse covariance matrix) estimator would affect the classification accuracy by proposing an analytic model on the upper bound of an LDA misclassification rate. Guided by the model, we propose a novel classifier, $\mathcal {DBSDA}$ , which improves classification accuracy through debiasing. Theoretical analysis shows that $\mathcal {DBSDA}$ possesses a reduced upper bound of misclassification rate and better asymptotic properties than sparse LDA (SDA). We conduct experiments on both synthetic datasets and real application datasets to confirm the correctness of our theoretical analysis and demonstrate the superiority of $\mathcal {DBSDA}$ over LDA, SDA, and other downstream competitors under HDLSS settings. [ABSTRACT FROM AUTHOR]

Nonnegative matrix factorization (NMF) is an advanced method for nonnegative feature extraction, with widespread applications. However, the NMF solution often entails to solve a global optimization problem with a nonconvex objective function and nonnegativity constraints. This paper presents a collective neurodynamic optimization (CNO) approach to this challenging problem. The proposed collective neurodynamic system consists of a population of recurrent neural networks (RNNs) at the lower level and a particle swarm optimization (PSO) algorithm with wavelet mutation at the upper level. The RNNs act as search agents carrying out precise local searches according to their neurodynamics and initial conditions. The PSO algorithm coordinates and guides the RNNs with updated initial states toward global optimal solution(s). A wavelet mutation operator is added to enhance PSO exploration diversity. Through iterative interaction and improvement of the locally best solutions of RNNs and global best positions of the whole population, the population-based neurodynamic systems are almost sure able to achieve the global optimality for the NMF problem. It is proved that the convergence of the group-best state to the global optimal solution with probability one. The experimental results substantiate the efficacy and superiority of the CNO approach to bound-constrained global optimization with several benchmark nonconvex functions and NMF-based clustering with benchmark data sets in comparison with the state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

We analyze the learning problem of one-hidden-layer nonoverlapping convolutional neural networks with the rectified linear unit (ReLU) activation function from the perspective of model estimation. The training outputs are assumed to be generated by the neural network with the unknown ground-truth parameters plus some additive noise, and the objective is to estimate the model parameters by minimizing a nonconvex squared loss function of the training data. Assuming that the training set contains a finite number of samples generated from the Gaussian distribution, we prove that the accelerated gradient descent (GD) algorithm with a proper initialization converges to the ground-truth parameters (up to the noise level) with a linear rate even though the learning problem is nonconvex. Moreover, the convergence rate is proved to be faster than the vanilla GD. The initialization can be achieved by the existing tensor initialization method. In contrast to the existing works that assume an infinite number of samples, we theoretically establish the sample complexity of the required number of training samples. Although the neural network considered here is not deep, this is the first work to show that accelerated GD algorithms can find the global optimizer of the nonconvex learning problem of neural networks. This is also the first work that characterizes the sample complexity of gradient-based methods in learning convolutional neural networks with the nonsmooth ReLU activation function. This work also provides the tightest bound so far of the estimation error with respect to the output noise. [ABSTRACT FROM AUTHOR]