Minimal Feedforward Parity Networks Using Threshold Gates

Fung, Hon-Kwok; Li, Leong Kwan

doi:10.1162/089976601300014556

Cited by 9 publications

(6 citation statements)

References 4 publications

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“…Indeed, it was reported in [22] that the most optimistic estimation for the number of the hidden neurons for the implementation of the n bit parity function using one hidden layer is √ n, (which is true for n ≤ 4), meanwhile the realistic estimation is O(n). In [14] it was shown up to n = 4, that the minimum size of the hidden layer required to solve the n-bit parity is n. It was theoretically estimated in [31] that the parity n function could be implemented using MLF with only (n + 1)/2 hidden neurons for n odd. However, this estimation was experimentally confirmed in [32] and [31] for maximum n = 7.…”

Section: Parity N Functionmentioning

confidence: 99%

Multilayer Feedforward Neural Network Based on Multi-valued Neurons (MLMVN) and a Backpropagation Learning Algorithm

2006

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A multilayer neural network based on multi-valued neurons (MLMVN) is considered in the paper. A multi-valued neuron (MVN) is based on the principles of multiple-valued threshold logic over the field of the complex numbers. The most important properties of MVN are: the complexvalued weights, inputs and output coded by the kth roots of unity and the activation function, which maps the complex plane into the unit circle. MVN learning is reduced to the movement along the unit circle, it is based on a simple linear error correction rule and it does not require a derivative. It is shown that using a traditional architecture of multilayer feedforward neural network (MLF) and the high functionality of the MVN, it is possible to obtain a new powerful neural network. Its training does not require a derivative of the activation function and its functionality is higher than the functionality of MLF containing the same number of layers and neurons. These advantages of MLMVN are confirmed by testing using parity n, two spirals and "sonar" benchmarks and the Mackey-Glass time series prediction.

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Section: Parity N Functionmentioning

confidence: 99%

Multilayer Feedforward Neural Network Based on Multi-valued Neurons (MLMVN) and a Backpropagation Learning Algorithm

2006

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“…This result explains that sometimes restraining neurons play a special role in the construction and analysis of BNNs. To some extent, the research of this paper also expands the range of the conclusion in [23]. A conclusion is reached that n is an upper bound of the minimum number of hidden neurons for the n-bit parity problem in BNNs.…”

Section: Discussionmentioning

confidence: 57%

“…BNNs that have single hidden layer and adopt linearly separable structures are easy to implement by hardware. But the problem of how to make certain the minimum number of hidden neurons for the n-bit parity problem has remained unsolved [16][17][18][19][20][21][22][23][24]. Ref.…”

Section: Introductionmentioning

confidence: 99%

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

Yang

Wang

Huang

2011

Sci. China Inf. Sci.

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Binary neural networks (BNNs) have important value in many application areas. They adopt linearly separable structures, which are simple and easy to implement by hardware. For a BNN with single hidden layer, the problem of how to determine the upper bound of the number of hidden neurons has not been solved well and truly. This paper defines a special structure called most isolated samples (MIS) in the Boolean space. We prove that at least 2 n−1 hidden neurons are needed to express the MIS logical relationship in the Boolean space if the hidden neurons of a BNN and its output neuron form a structure of AND/OR logic. Then the paper points out that the n-bit parity problem is just equivalent to the MIS structure. Furthermore, by proposing a new concept of restraining neuron and using it in the hidden layer, we can reduce the number of hidden neurons to n. This result explains the important role of restraining neurons in some cases. Finally, on the basis of Hamming sphere and SP function, both the restraining neuron and the n-bit parity problem are given a clear logical meaning, and can be described by a series of logical expressions.

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“…These two examples one can find in any modern book on neural networks, as well as it is possible to find many specific ideas about design of the most efficient network for solving the Parity problem (see, for example Fung and Li 2001;Mizutani et al 2000). The number of linearly separable Boolean functions of n variables is very small in comparison with the number of all Boolean functions of n variables for n > 3.…”

Section: Introductionmentioning

confidence: 99%

Solving the XOR and parity N problems using a single universal binary neuron

Aizenberg

2007

Soft Comput

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A universal binary neuron (UBN) operates with complex-valued weights and a complex-valued activation function, which is the function of the argument of the weighted sum. The activation function of the UBN separates a whole complex plane onto equal sectors, where the activation function is equal to either 1 or −1 depending on the sector parity (even or odd, respectively). Thus, the UBN output is determined by the argument of the weighted sum. This makes it possible the implementation of the nonlinearly separable (non-threshold) Boolean functions on a single neuron. Hence, the functionality of UBN is incompatibly higher than the functionality of the traditional perceptron. In this paper, we will consider a new modified learning algorithm for the UBN. We will show that classical nonlinearly separable problems XOR and Parity n can be easily solved using a single UBN, without any network. Finally, it will be considered how some other important nonlinearly separable problems may be solved using a single UBN.

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Minimal Feedforward Parity Networks Using Threshold Gates

Cited by 9 publications

References 4 publications

Multilayer Feedforward Neural Network Based on Multi-valued Neurons (MLMVN) and a Backpropagation Learning Algorithm

Multilayer Feedforward Neural Network Based on Multi-valued Neurons (MLMVN) and a Backpropagation Learning Algorithm

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

Solving the XOR and parity N problems using a single universal binary neuron

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