DOI QR코드

DOI QR Code

The Implementable Functions of the CoreNet of a Multi-Valued Single Neuron Network

단층 코어넷 다단입력 인공신경망회로의 함수에 관한 구현가능 연구

  • Park, Jong Joon (Dept. of Computer Science, College of Natural Science & Engineering, Seokyeong University)
  • Received : 2014.11.17
  • Accepted : 2014.12.24
  • Published : 2014.12.31

Abstract

One of the purposes of an artificial neural netowrk(ANNet) is to implement the largest number of functions as possible with the smallest number of nodes and layers. This paper presents a CoreNet which has a multi-leveled input value and a multi-leveled output value with a 2-layered ANNet, which is the basic structure of an ANNet. I have suggested an equation for calculating the capacity of the CoreNet, which has a p-leveled input and a q-leveled output, as $a_{p,q}={\frac{1}{2}}p(p-1)q^2-{\frac{1}{2}}(p-2)(3p-1)q+(p-1)(p-2)$. I've applied this CoreNet into the simulation model 1(5)-1(6), which has 5 levels of an input and 6 levels of an output with no hidden layers. The simulation result of this model gives, the maximum 219 convergences for the number of implementable functions using the cot(${\sqrt{x}}$) input leveling method. I have also shown that, the 27 functions are implementable by the calculation of weight values(w, ${\theta}$) with the multi-threshold lines in the weight space, which are diverged in the simulation results. Therefore the 246 functions are implementable in the 1(5)-1(6) model, and this coincides with the value from the above eqution $a_{5,6}(=246)$. I also show the implementable function numbering method in the weight space.

인공신경망회로 목표 중의 하나는 최소한의 회로구성으로 구현가능함수를 가능한 많게 하는데 있다. 본 논문은 인공신경망회로의 가장 기본이 되는 하나의 입력노드와 하나의 출력노드, 그리고 입출력에 다단(multi-level)값을 갖는 단층(입출력 2 layer) 다단 코어넷(CoreNet)을 제안하고 그 처리 용량을 구하였고, 무게값 공간에서 구현 가능한 함수와 각 무게값 좌표(${\omega}$,${\theta}$)를 계산으로 구하여 한 함수의 구현 가능 여부를 알 수 있게 하였다. 또 입력 단계(level)값 설정 방법으로 cot(${\sqrt{x}}$)을 제안하였다. 제안된 p단 입력과 q단 출력을 갖는 코어넷의 처리용량(구현 가능한 함수의 수)은 $a_{p,q}={\frac{1}{2}}p(p-1)q^2-{\frac{1}{2}}(p-2)(3p-1)q+(p-1)(p-2)$임을 유도 증명하였다. 시뮬레이션으로 5단(level) 입력 값과, 6단 출력 값을 갖는 1(5)-1(6) 모델을 분석한 결과, cot(${\sqrt{x}}$) 입력 레벨링법에서 총 246가지의 함수가 구현가능 함을 보였다. 이 모델의 시뮬레이션 결과에서는 최대 219개의 함수가 수렴(구현 가능)하였고, 구현가능 함수 중에서 나머지 수렴되지 않은 27개의 함수는 무게값 공간에서 무게값 좌표를 계산하여 구현 가능함을 보였다. 이는 앞에서 제시된 코어넷 처리용량 $a_{5,6}(=246)$에 의한 계산 값과 일치하였다. 무게값 공간에서, 구현 가능한 함수가 차지하는 영역의 함수번호 매김 방법도 제시하여 구현 가능함수의 번호도 알 수 있도록 하였다.

Keywords

References

  1. Takahiro Haga, "An application of the (p,q)-logic to the synthesis of the p-valued logical networks and the s-(p,q)-logical completeness" Information Sciences 115, pp.165-185, 1999 https://doi.org/10.1016/S0020-0255(98)10075-0
  2. Igor Aizenberg, "Solving The XOR and Parity n Problems Using a Single Universal Binary Neuron", Soft Computing, vol.12, No3, 2008.2
  3. Igor Aizenberg, Shane Alexander, and Jacob Jackson "Recognition of Blurred Images Using Multilayer Neural Network Based on Multi-valued Neurons", 2011 41st IEEE International Symposium on Multiple-Valued Logic. 2011
  4. Igor Aizenberg, Claudio Moraga, and Dmitry Paliy, "A Feedforward Neural Network based on Multi-Valued Neurons", Computational Intelligence, Theory and Applications. Advances in Soft Computing, XIV, Springer, 2005
  5. Jong Joon Park, "The Capacity of Multi-Valued single layer CoreNet(Neural network) and Precalculation of its Weight Values", Journal of IKEEE, Vol.15 No4, pp.354-362, 2011.12
  6. Jong Joon Park, "The Functions of Core-Net with 3-levels 2-layered Artificial Neural Netoworks", Institute of Industrial Technology, Seokyeong University, 25, pp.1-11, 2010
  7. Jong Joon Park, "The Leveling Method for the Input Values of a Multi-valued Core-net Simulation", Institute of Industrial Technology, Seokyeong University, 26, pp.69-80, 2011.06
  8. Jong Joon Park, Kandel, Abraham, Langholz, G. and L. Hawkes, "Neural Network Processing of Linguistic Symbols, Fuzzy Sets, Neural Networks, and Soft Computing", Ed. by R.R. Yager and L.A.Zadeh, Van Nostrand Reinhold, pp.265-284, 1994.
  9. Jong Joon Park, "The Capacity of CoreNet: Multi-Level 2-Layer Neural Networks", The KIPS Transactions, Vol.6, No.8, pp.2098-2105, 1999
  10. Harding, E.F., "The number of partitions of a set of N points in k dimensions induced. by hyperplanes", Proceedings of the Edinburgh Mathematical Society, Series II 15, 285-289, 1967.7
  11. Takahisa. Toda, On Partitioning Colored Points, http://arxiv.org/abs/1011.3451v1, http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.3451v1.pdf, pp.2, 2010,10 https://doi.org/10.1587/transfun.E94.A.1242
  12. Hirose, Yoshio, Yamashita, Koichi, and Hijiya, Shimpei, "Back-Propagation Algorithm Which Varies the Number of Hidden Units", Neural Networks, Vol.4, pp.61-66, 1991 https://doi.org/10.1016/0893-6080(91)90032-Z
  13. Igor Aizenberg, "Learning Nonlinearly Separable mod k Addition Problem Using a Single Multi-Valued Neuron With a Periodic Activation Function", WCCI 2010 IEEE World congress on Computational Intelligence, pp.13-23, 2010-CCIB, Barcelona, Spain