Research Article
Single Neuron for Solving XOR like Nonlinear Problems
Ashutosh Mishra , Jaekwang Cha , and Shiho Kim
School of Integrated Technology, YICT, Yonsei University, Seoul, Republic of Korea
Correspondence should be addressed to Shiho Kim; shiho@yonsei.ac.kr
Received 16 October 2021; Revised 24 March 2022; Accepted 13 April 2022; Published 28 April 2022
Academic Editor: Jos´ e Alfredo Hern´ andez-P´ erez
Copyright © 2022 Ashutosh Mishra et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
XOR is a special nonlinear problem in artificial intelligence (AI) that resembles multiple real-world nonlinear data distributions. A
multiplicative neuron model can solve these problems. However, the multiplicative model has the indigenous problem of
backpropagation for densely distributed XOR problems and higher dimensional parity problems. To overcome this issue, we have
proposed an enhanced translated multiplicative single neuron model. It can provide desired tessellation surface. We have
considered an adaptable scaling factor associated with each input in our proposed model. It helps in achieving optimal scaling
factor value for higher dimensional input. e efficacy of the proposed model has been tested by randomly increasing input
dimensions for XOR-type data distribution. e proposed model has crisply classified even higher dimensional input in their
respective class. Also, the computational complexity is the same as that of the previous multiplicative neuron model. It has shown
more than an 80% reduction in absolute loss as compared to the previous neuron model in similar experimental conditions.
erefore, it can be considered as a generalized artificial model (single neuron) with the capability of solving XOR-like
real problems.
1. Introduction
Minski and Perpert deduced that the XOR problem requires
more than one hyperplane [1]. ey provide a more gen-
eralized artificial neuron model by introducing the concept
of weights and proved the inability of a single perceptron for
solving ‘Exclusive-OR (XOR)’ [2]. e XOR problem is
symmetrical to other popular and real-world problems such
as XOR type nonlinear data distribution in two classes, N-bit
parity problems. [3]. erefore, many researchers tried to
find a suitable way out to solve the XOR problem [4–15].
Although, most of the solutions are for the classical XOR
problem. ey either use more than one layer or provide a
complex solution for two-bit logical XOR only. Few of these
used the complex value neuron model, eventually creating
one more layer (i.e., hidden layer). Because the complex
value neuron model requires representing the real input in a
complex domain, one approach is based on the multipli-
cative neuron model. is is translated multiplicative neuron
(π
t
-neuron) approach [16, 17]. ey have modified the
π-neuron model (which generates the decision surfaces
centered at the origin of input) to an extended multiplicative
neuron, i.e., a π
t
-neuron model for solving the N-bit parity
problems by creating tessellation surfaces. However, it has
limitations for higher dimensional N-bit parity problems. It
is suitable for up to six dimensions. For seven and higher
dimensional inputs, it has reported poor accuracy [17]. In
other words, it has a convergence problem for higher di-
mensional inputs. It is merely because of the multiplicative
nature of the model. More clearly, the infinitesimal errors in
the model obtain a much smaller value after getting mul-
tiplied in case of higher dimensional inputs, consequently
vanishing the gradient. erefore, a convergence problem
occurs in this model for higher-dimensional inputs.
To overcome the issue of the π
t
-neuron model, we have
proposed an enhanced translated multiplicative model
neuron (π
t
-neuron) model in this paper. It helps in achieving
mutually orthogonal separation in the case of two-bit
classical XOR data distribution. Also, the proposed model
has shown the capability for solving the higher-order N-bit
parity problems. erefore, it is a generalized artificial model
for solving real XOR problems. To examine this claim, we
Hindawi
Computational Intelligence and Neuroscience
Volume 2022, Article ID 9097868, 11 pages
https://doi.org/10.1155/2022/9097868