张量型二阶随机向量函数链接网络

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"基于张量的类型-2随机向量功能链接网络" 在当前研究中，作者Guoliang Zhao和Wei Wu提出了一种扩展版的随机向量功能链接（RVFL）网络，通过引入张量结构并用类型-2模糊集替换增强节点。这种创新的方法分为两个版本：区间类型-2 RVFL网络和张量基类型-2 RVFL网络。传统随机向量功能链接网络是一种神经网络模型，它增加了非线性映射能力，通过额外的增强节点实现。在区间类型-2 RVFL网络中，虽然基本架构与原始RVFL网络相似，但关键区别在于增强节点的输出不再是简单的权重值，而是通过不确定性权重方法进行去模糊化的结果。这反映了输入数据的不确定性和模糊性，增强了模型对复杂数据的处理能力。另一方面，张量基类型-2 RVFL网络更进一步，不仅考虑了去模糊化的结果，还结合了下隶属函数和上隶属函数的值。这些值被构造成一个四维张量，该张量由三个组件组成。随后，利用这个张量的Moore-Penrose逆来处理和分析数据，这提高了模型的计算效率和表达能力。张量在数学中是一种多维数组，它在处理复杂数据结构，如图像、视频和高阶关系数据时非常有用。在这里，张量被用来捕获和表示类型-2模糊集中的复杂信息，从而提高网络的泛化能力和适应性。通过这种方式，网络可以更好地理解和学习非结构化和模糊的数据模式，这在许多实际应用中，如图像识别、语音处理和大数据分析中都极其重要。此外，Moore-Penrose逆是一种广义逆矩阵，用于处理不具备方阵或满秩的矩阵问题。在网络中，它可以用于解决张量运算中的线性方程组，确保了模型的稳定性和计算的准确性。总结来说，"基于张量的类型-2随机向量功能链接网络"是一种结合了模糊理论和张量计算的新型神经网络模型，旨在提升对不确定性和复杂数据的建模能力。通过这样的方法，研究者们期望能够开发出更加智能和适应性强的机器学习系统，以应对现实世界中多样化的挑战。

Tensor-based type-2 random vector functional link network

Guoliang Zhao

1,2

,WeiWu

1. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Peoples Republic of China

E-mail: guoliangzhao@imu.edu.cn

2. College of Electronic Information Engineering, Inner Mongolia University, Hohhot 010021, Peoples Republic of China

Abstract: In the paper, random vector functional link (RVFL) network is extended by the tensor structure,

and the enhancement nodes are replaced by type-2 fuzzy sets. Two versions of RVFL network are proposed,

one is the interval type-2 RVFL network, and the other is the tensor-based type-2 RVFL network. The interval

type-2 RVFL network is similar to the original RVFL, the only difference is that the outputs of enhancement

nodes are the defuzziﬁcation results of the uncertainty weight method. The tensor-based type-2 RVFL net-

work adopts the defuzziﬁcation results of the uncertainty weight method, together with values of the lower

membership functions and values of upper membership functions, a 4-dimensional tensor is then formed

based on the three components; then, Moore-Penrose inverse of even-order tensor which is implemented via

Einstein product is used for solving the tensor equation. Moreover, the weighting vector is obtained via a

balance weighting factor, which is resulted from a weighted average of the weighting vector of RVFL network

and the weighting vector of the solution to the abovementioned tensor equation. Finally, the proposed algo-

rithms are veriﬁed on three nonlinear benchmark functions, a nonlinear system identiﬁcation problem, and

four regression problems.

Key Words: Functional-link network (FLN), Random vector functional link (RVFL) network, Tensor equation,

Interval type-2 RVFL (IT2-RVFL), Tensor-based type-2 RVFL (TT2-RVFL)

1 INTRODUCTION

FLNs are single-layered neural networks that impose

nonlinearity on input layer using many kinds of non-

linear functions [1]. FLNs can be categorized into two

groups based on the type of weighting factors: RVFL

network and functional-based FLN depending on the

usage of various types of function basis. With ran-

domly generated weights between input and hidden

layers, RVFL network is a universal approximator for

continuous functions on compact sets with fast learn-

ing property [2]. RVFL network exploits connection

between the input layer and the output layer through

orthogonal functions [3]. The orthogonal polynomial-

based RVFL network is proposed by utilizing advan-

tages from expansion of the input vector and ran-

dom determination of the input weights. A parsimo-

nious RVFL is proposed [4] to endorse the network

with self-organizing property. Furthermore, FLN is

also modiﬁed and trained by artiﬁcial bee colony,

and the drawback of BP-learning scheme is overcame

by the intelligent algorithm [5]. Moreover, the ef-

fect of input-output connections of the solar power

data of Sydney is studied, and it is found out that

RVFL performs better than other random weight sin-

gle shrouded layer feed-forward neural networks and

single shrouded layer feed-forward neural networks

[6]. Based on the learning strategy that uses privi-

This work is supported by National Nature Science Foundation

of China under Grant 61603126 and 61773088.

leged information paradigm, an alternative way to

train RVFL networks is provided, which is named

as RVFL+ network [7]. An RVFL network ensemble

is introduced by employing ”perturb and combine”

strategies, and the proposed RVFL network ensem-

ble also outperforms all neural network-based meth-

ods used in the experiments [8]. To serve as general

guidelines for designing RVFL networks-based clas-

siﬁers, Zhang and Suganthan [9] studied ﬁve factors,

that is, output layer’s bias, direct links, activation

function type, scaling of parameter and the solution

procedure of the RVFL, which effect the classiﬁcation

ability of RVFL networks.

To the FLNs, a functional link-based novel recur-

rent fuzzy neural network is applied for the static

synchronous compensator to reduce the power ﬂuc-

tuations, the proposed controller can achieve better

damping characteristics and effectively stabilize the

network under unstable conditions [10]. A func-

tional link-based recurrent FNN for the variable-

speed switched reluctance generator control has been

studied by Hong and Huang, and the node connect-

ing weights of the net are trained online by back-

propagation algorithms [11]. Functional link least

square principle has been used to solve robust regres-

sion problems [12]. An interactively recurrent self-

evolving FNN which employs an FLN to the con-

sequent part of fuzzy rules is proposed for predic-

tion and identiﬁcation of dynamic systems [13], and

the mapping ability and the performance results of

3278

978-1-7281-0106-4/19/$31.00

2019 IEEE

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