模糊受限玻尔兹曼机增强深度学习

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"Fuzzy Restricted Boltzmann Machine for Enhancing Deep Learning" 本文主要探讨了模糊受限玻尔兹曼机（Fuzzy Restricted Boltzmann Machine, FRBM）在深度学习中的应用和增强作用。近年来，深度学习在机器学习领域引发了研究热潮，并在模式识别、图像识别、语音识别以及视频处理等多个领域展现出卓越的性能。受限玻尔兹曼机（Restricted Boltzmann Machine, RBM）是当前深度学习技术中的关键组件，许多现有的深度网络都是基于或与RBM相关的。传统的RBM是一种二元概率模型，其结构包括可见层和隐藏层，两层之间存在相互连接但层内无连接的权重。RBM通过能量函数定义了其概率分布，并利用 Contrastive Divergence 等算法进行参数训练，从而能捕获数据的潜在特征。然而，常规RBM对数据的表示往往较为简单，难以处理具有模糊性和不确定性的复杂数据。模糊RBM引入了模糊理论，允许单元状态不再是简单的二进制，而是可以有连续的隶属度，这使得模型能够更好地适应和表达模糊信息。模糊逻辑的应用增强了RBM处理非线性关系和不确定性数据的能力，提高了模型的泛化性能。FRBM的权重和激活函数可以设计成模糊逻辑的形式，如隶属函数，以适应不同类型的模糊信息。文章可能详细讨论了FRBM的建模过程，包括如何定义模糊成员函数、如何调整模糊规则以及如何优化训练过程以适应模糊环境。作者可能还对比了FRBM与传统RBM在深度学习架构中的表现，展示了在某些任务中模糊模型的优越性。此外，文章可能会介绍一些实际应用案例，证明FRBM在处理模糊或复杂数据时的有效性和优势。在实验部分，可能通过一系列的基准测试和对比实验，评估了FRBM在图像分类、语音识别等任务上的性能，并与传统的深度学习方法进行了比较。实验结果可能表明，FRBM在处理模糊信息和复杂数据集时，能够提高准确率和鲁棒性，验证了其在深度学习框架中的增强作用。这篇文章为深度学习提供了一个新的视角，即通过模糊理论改进受限玻尔兹曼机，以提升模型对模糊数据的处理能力。这不仅扩展了RBM的理论基础，也为解决现实世界中充满不确定性的复杂问题提供了新的工具。对于那些涉及模糊信息处理的深度学习应用，如自然语言处理、计算机视觉等，FRBM可能成为一种强大的解决方案。

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI

10.1109/TFUZZ.2015.2406889, IEEE Transactions on Fuzzy Systems

IEEE TRANSACTIONS ON FUZZY SYSTEMS 3

III. FUZZY RESTRICTED BOLTZMANN MACHINE AND ITS

LEARNING ALGORITHM

A. Fuzzy Restricted Boltzmann Machine

The proposed novel fuzzy restricted Boltzmann machine

(FRBM) is illustrated in Fig. 3, in which the connection

weighs and biases are fuzzy parameters denoted by

θ. There

are several merits of the FRBM model. The ﬁrst one is that

the FRBM has much better representation than the regular

RBM in modelling probabilities over visible and hidden units.

Speciﬁcally, the RBM is only a special case of the FRBM

when no fuzziness exists in the FRBM model. The second

one is that the robustness of the FRBM model surpasses RBM

model. The FRBM shows out more robustness when it comes

to the ﬁtting of the model with noisy data. All these advantages

spring from the fuzzy extension of the relationships between

cross-layer variables, and inherit the characteristics of fuzzy

models.

Fig. 3: Fuzzy Restricted Boltzmann Machine

Since the FRBM is an extension of the RBM model, so the

discussion starts from a brief introduction on the RBM model.

An RBM is an energy-based probabilistic model, in which the

probability distribution is deﬁned through an energy function.

Its probability is deﬁned as

P (x, h, θ) =

−E(x,h,θ)

, (7)

Z =

˜x

−E(˜x,

h,θ)

, (8)

where E(x, h, θ) is the energy function, θ are the parameters

governing the model, Z is the normalizing factor which is

called the partition function,

x and

h are two vector variables

representing visible and hidden units that are used to traverse

and summarize all the conﬁgurations of units on the graph.

The energy function for the RBM is deﬁned by

E(x, h, θ) = −b

x − c

h − h

Wx, (9)

where b

and c

are the offsets, and W

is the connection

weight between j-th visible unit and i-th hidden unit, and θ =

{b, c, W}.

To establish fuzzy restricted Boltzmann machine, it is nec-

essary to ﬁrstly deﬁne the fuzzy energy function for the model.

The fuzzy energy function can be extended from Eqn. (9) in

accordance with extension principle as follow

E(x, h, θ) = −b

x − c

h − h

Wx, (10)

where

E(x, h, θ) is a fuzziﬁed energy function, and θ =

{b, c, W} are fuzzy parameters. Correspondingly, the fuzzy

free energy

F, which marginalize hidden units and map Eqn.

(7) into a simpler one, is deduced as

F(x, θ) = − log

−

E(x,

h,θ)

, (11)

where

F is extended from crisp free energy function F:

F(x, θ) = − log

−E(x,

h,θ)

. (12)

If the fuzzy free energy function is directly employed to

deﬁne the probability, it leads to a fuzzy probability [27].

Finally, the optimization in learning process turns into a fuzzy

maximum likelihood problem. However, this kind of problem

is quite intractable because the fuzzy objective function is non-

linear and the membership function is difﬁcult to compute,

since the computation of its alpha-cuts become NP-hard prob-

lems [29]. Therefore, it is necessary to transform the problem

into regular maximum likelihood problem by defuzzifying the

fuzzy free energy function (11). The centre of area (centroid)

method [30] is employed to defuzzify the fuzzy free energy

function

F(x). Then the likelihood function can be deﬁned by

the defuzziﬁed fuzzy free energy function. Consequently, the

fuzzy optimization problem becomes real-valued problem, and

conventional optimization approaches can be directly applied

to ﬁnd the optimal solutions. The centroid of fuzzy number

F(x) is denoted by F

(x), and has the following form

(x,

θ) =

θF(x, θ)dθ

F(x, θ)dθ

, θ ∈

θ. (13)

Naturally, after the fuzzy free energy is defuzziﬁed, the prob-

ability can be deﬁned as

(x,

θ) =

−F

(x;

θ)

, Z =

˜x

−F

(˜x,

θ)

. (14)

In the fuzzy RBM model, the objective function is the negative

log-likelihood, which is given by

θ, D) = −

x∈D

log P

(x,

θ), (15)

where D is the training dataset.

The learning problem is to ﬁnd optimal solutions for param-

eters

θ that minimize the objective function L(θ, D), i.e.,

min

θ, D) (16)

In the following subsection, the detailed procedure to address

the dual problem of maximum likelihood by utilizing stochas-

tic gradient descent method will be investigated.

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