The upper bound of the minimal number of hidden neurons for the parity problem in binary neural networks

Yang, Juan; Wang, Qiang; Huang, Zhensheng

doi:10.1007/s11432-011-4405-6

Cited by 6 publications

(3 citation statements)

References 20 publications

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“…Then, the number of nodes was gradually increased until the learning error was no longer considerably reduced. The optimal number of nodes in a hidden layer is commonly determined using Equation 1 [38,41,42].…”

Section: Topology Of the Bp Neural Network With Double Hidden Layersmentioning

confidence: 99%

Groundwater Level Prediction for the Arid Oasis of Northwest China Based on the Artificial Bee Colony Algorithm and a Back-propagation Neural Network with Double Hidden Layers

Zheng

et al. 2019

Water

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Groundwater is crucial for economic and agricultural development, particularly in arid areas where surface water resources are extremely scarce. The prediction of groundwater levels is essential for understanding groundwater dynamics and providing scientific guidance for the rational utilization of groundwater resources. A back propagation (BP) neural network based on the artificial bee colony (ABC) optimization algorithm was established in this study to accurately predict groundwater levels in the overexploited arid areas of Northwest China. Recharge, exploitation, rainfall, and evaporation were used as input factors, whereas groundwater level was used as the output factor. Results showed that the fitting accuracy, convergence rate, and stabilization of the ABC-BP model are better than those of the particle swarm optimization (PSO-BP), genetic algorithm (GA-BP), and BP models, thereby proving that the ABC-BP model can be a new method for predicting groundwater levels. The ABC-BP model with double hidden layers and a topology structure of 4-7-3-1, which overcame the overfitting problem, was developed to predict groundwater levels in Yaoba Oasis from 2019 to 2030. The prediction results of different mining regimes showed that the groundwater level in the study area will gradually decrease as exploitation quantity increases and then undergo a decline stage given the existing mining condition of 40 million m3/year. According to the simulation results under different scenarios, the most appropriate amount of groundwater exploitation should be maintained at 31 million m3/year to promote the sustainable development of groundwater resources in Yaoba Oasis.

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Section: Topology Of the Bp Neural Network With Double Hidden Layersmentioning

confidence: 99%

Groundwater Level Prediction for the Arid Oasis of Northwest China Based on the Artificial Bee Colony Algorithm and a Back-propagation Neural Network with Double Hidden Layers

Zheng

et al. 2019

Water

View full text Add to dashboard Cite

show abstract

“…Various efforts have been made to explore the relations between the approximation ability and the number of nodes of some specific neural network, such as singlehidden-layer feedforward neural networks (SLFNs), and two-hidden-layer feedforward neural networks with specific or conditional activation functions [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. For example, it was proved that N arbitrary distinct samples can be learned precisely by standard SLFNs with N hidden neurons (including biases) and the signum activation function in [12].…”

Section: Introductionmentioning

confidence: 99%

Upper bounds on the node numbers of hidden layers in MLPs

Liu¹,

Feng²,

Du³

et al. 2021

NNW

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It is one of the fundamental and challenging problems to determine the node numbers of hidden layers in neural networks. Various efforts have been made to study the relations between the approximation ability and the number of hidden nodes of some specific neural networks, such as single-hidden-layer and two-hiddenlayer feedforward neural networks with specific or conditional activation functions. However, for arbitrary feedforward neural networks, there are few theoretical results on such issues. This paper gives an upper bound on the node number of each hidden layer for the most general feedforward neural networks called multilayer perceptrons (MLP), from an algebraic point of view. First, we put forward the method of expansion linear spaces to investigate the algebraic structure and properties of the outputs of MLPs. Then it is proved that given k distinct training samples, for any MLP with k nodes in each hidden layer, if a certain optimization problem has solutions, the approximation error keeps invariant with adding nodes to hidden layers. Furthermore, it is shown that for any MLP whose activation function for the output layer is bounded on R, at most k hidden nodes in each hidden layer are needed to learn k training samples.

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“…However, from a theoretical view, problems of the lower and upper bounds of neurons in hidden layers and the bounds of hidden layers to classification and pattern recognition architectures have not been completely studied yet. For instance, one of the current existing findings is related to binary neural networks [18]. They defined a structure named a most isolated samples in the Boolean field and proved that at least 1 2 n hidden computing@computingonline.net www.computingonline.net…”

Section: Introductionmentioning

confidence: 99%

Approach to the Synthesis of Neural Network Structure During Classification

Marakhimov¹,

Khudaybergenov²

2020

IJC

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Evaluating the number of hidden neurons necessary for solving of pattern recognition and classification tasks is one of the key problems in artificial neural networks. Multilayer perceptron is the most useful artificial neural network to estimate the functional structure in classification. In this paper, we show that artificial neural network with a two hidden layer feed forward neural network with d inputs, d neurons in the first hidden layer, 2d+2 neurons in the second hidden layer, k outputs and with a sigmoidal infinitely differentiable function can solve classification and pattern problems with arbitrary accuracy. This result can be applied to design pattern recognition and classification models with optimal structure in the number of hidden neurons and hidden layers. The experimental results over well-known benchmark datasets show that the convergence and the accuracy of the proposed model of artificial neural network are acceptable. Findings in this paper are experimentally analyzed on four different datasets from machine learning repository.

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The upper bound of the minimal number of hidden neurons for the parity problem in binary neural networks

Cited by 6 publications

References 20 publications

Groundwater Level Prediction for the Arid Oasis of Northwest China Based on the Artificial Bee Colony Algorithm and a Back-propagation Neural Network with Double Hidden Layers

Groundwater Level Prediction for the Arid Oasis of Northwest China Based on the Artificial Bee Colony Algorithm and a Back-propagation Neural Network with Double Hidden Layers

Upper bounds on the node numbers of hidden layers in MLPs

Approach to the Synthesis of Neural Network Structure During Classification

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