Computing convex hull is one of the central problems账in various applications of computational geometry. In this paper, a convex hull computing neural network (CHCNN) is developed to solve the related problems in the N-dimensional spaces. The algorithm is based on a two-layered neural network, topologically similar to ART, with a newly developed adaptive training strategy called excited learning. The CHCNN provides a parallel on-line and real-time processing of data which, after training, yields two closely related approximations, one from within and one from outside, of the desired convex hull. It is shown that accuracy of the approximate convex hulls obtained is around O[K-1=(N-1)], where K is the number of neurons in the output layer of the CHCNN. When K is taken to be sufficiently large, the CHCNN can generate any accurate approximate convex hull. We also show that an upper bound exists such that the CHCNN will yield the precise convex hull when K is larger than or equal to this bound. A series of simulations and applications is provided to demonstrate the feasibility, effectiveness, and high efficiency of the proposed algorithm.

In this paper we propose an L1/2 regularizer which has a nonconvex penalty. The L1/2 regularizer is shown to have many promising properties such as unbiasedness, sparsity and Oracle properties. A reweighed iterative algorithm is proposed so that the solution of the L1/2 regularizer can be solved through transforming it into the solution of a series of L1 regularizers. The solution of the L1/2 regularizer is more sparse than that of the L1 regularizer, while solving the L1/2 regularizer ismuch simpler than solving the L0 regularizer. The experiments show that the L1/2 regularizer is very useful and efficient, and can be taken as a representative of the Lp(0<p<1) regularizer.

The generalization performance is the main concern of machine learning theoretical research. The previous main bounds describing the generalization ability of the Empirical Risk Minimization (ERM) algorithm are based on independent and identically distributed (i.i.d.) samples. In order to study the generalization performance of the ERM algorithm with dependent observations, we first establish the exponential bound on the rate of relative uniform convergence of the ERM algorithm with exponentially strongly mixing observations, and then we obtain the generalization bounds and prove that the ERM algorithm with exponentially strongly mixing observations is consistent. The main results obtained in this paper not only extend the previously known results for i.i.d. observations to the case of exponentially strongly mixing observations, but also improve the previous results for strongly mixing samples. Because the ERM algorithm is usually very time-consuming and overfitting may happen when the complexity of the hypothesis space is high, as an application of our main results we also explore a new strategy to implement the ERM algorithm in high complexity hypothesis space.

For the nearly exponential type of feedforward neural networks (neFNNs), it is revealed the essential order of their approximation. It is proven that for any continuous function defined on a compact set of Rd, there exists a three-layer neFNNs with fixed number of hidden neurons that attain the essential order. When the function to be approximated belongs to the α-Lipschitz family (0<α≤2), the essential order of approximation is shown to be O(n-α) where n is any integer not less than the reciprocal of the predetermined approximation error. The upper bound and lower bound estimations on approximation precision of the neFNNs are provided. The obtained results not only characterize the intrinsic property of approximation of the neFNNs, but also uncover the implicit relationship between the precision (speed) and the number of hidden neurons of the neFNNs.

The neuron state modeling and the local field modeling provides two fundamental modeling approaches to neural network research, based on which a neural network system can be called either as a static neural network model or as a local field neural network model. These two models are theoretically compared in terms of their trajectory transformation property, equilibrium correspondence property, nontrivial attractive manifold property, global convergence as well as stability in many different senses. The comparison reveals an important stability invariance property of the two models in the sense that the stability (in any sense) of the static model is equivalent to that of a subsystem deduced from the local field model when restricted to a specific manifold. Such stability invariance property lays a sound theoretical foundation of validity of a useful, cross-fertilization type stability analysis methodology for various neural network models.