Sussner and Valle's models work quite well in auto-associative mode with perfect input patterns, similar to other improvements of Kosko's model.. Depending on the ratio of association an
Trang 1Improving learning rule for fuzzy associative memory
with combination of content and association
Human Machine Interaction Laboratory, University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam
a r t i c l e i n f o
Article history:
Received 4 July 2013
Received in revised form
14 September 2013
Accepted 8 January 2014
Available online 4 August 2014
Keywords:
Fuzzy associative memory
Noise tolerance
Pattern associations
a b s t r a c t
FAM is an associative memory that uses operators of fuzzy logic and mathematical morphology (MM) FAMs possess important advantages including noise tolerance, unlimited storage, and one pass convergence An important property, deciding FAM performance, is the ability to capture content of each pattern, and association of patterns Existing FAMs capture either content or association of patterns well, but not both of them They are designed to handle either erosive or dilative noise in distorted inputs but not both Therefore, they cannot recall distorted input patterns very well when both erosive and dilative noises are present In this paper, we propose a new FAM called content-association associative memory (ACAM) that stores both content and association of patterns The weight matrix is formed with the weighted sum of output pattern and the difference between input and output patterns Our ACAM can handle inputs with both erosive and dilative noises better than existing models
& 2014 Elsevier B.V All rights reserved
1 Introduction
Associative memories (AMs) store pattern associations and
can retrieve desired output pattern upon presentation of a
possibly noisy or incomplete version of an input pattern They
are categorized as auto-associative memories and
hetero-associative memories A memory is said to be auto-hetero-associative
if the output is the same as the input On the other hand, the
memory is considered hetero-associative if the output is
differ-ent from the input The Hopfield network [1] is probably the
most widely known auto-associative memory at present with
many variations and generalizations Among different kinds of
associative memories, fuzzy associative memories (FAMs)
belong to the class of fuzzy neural networks, which combine
fuzzy concepts and fuzzy inference rules with the architecture
and learning of neural networks Input patterns, output
pat-terns, and/or connection weights of FAMs are fuzzy-valued
Working with uncertain data is the reason why FAMs have been
used in manyfields such as pattern recognition, control,
estima-tion, inference, and prediction For example, Sussner and Valle
used the implicative FAMs for face recognition[2] Kim et al
predicted Korea stock price index[3] Shahir and Chen inspected
the quality of soaps on-line [4] Wang and Valle detected
pedestrian abnormal behaviour[5] Sussner and Valle predicted
the Furnas reservoir from 1991 to 1998[2]
Kosko's FAM [6]in the early 1990s has initiated research on FAMs For each pair of input X and output Y, Kosko's FAM stores their association as the fuzzy rule “If x is X then y is Y” in a separated weight matrix called FAM matrix Thus, Kosko's overall fuzzy system comprises several FAM matrices Therefore, the disadvantage of Kosko's FAM is very low storage capacity In order
to overcome this limitation, different improved FAM versions have been developed that store multiple pattern associations in a single FAM matrix[7–10] In Chung and Lee's model[7] which gener-alizes Kosko's one, FAM matrices are combined with a max-t composition into a single matrix It is shown that all outputs can
be recalled perfectly with the single combined matrix if the input patterns satisfy certain orthogonality conditions The fuzzy impli-cation operator is used to present associations by Junbo et al.[9], which improves the learning algorithm for Kosko's max–min FAM model By adding a threshold at recall phase, Liu has modi-fied Junbo's FAM in order to improve the storage capacity [10] Recently, Sussner and Valle has established implicative fuzzy associative memories (IFAMs)[2]with implicative fuzzy learning This can be considered as a class of associative memories that grew out of morphological associative memories[11]because each node performs a morphological operation Sussner and Valle's models work quite well in auto-associative mode with perfect input patterns, similar to other improvements of Kosko's model However, these models suffer much from the presence of both erosive and dilative noises
In binary mode, many associative memory models show their noise tolerance capability from distorted input based on their own mathematical characteristics [10,2] For example, models
Contents lists available atScienceDirect
journal homepage:www.elsevier.com/locate/neucom
Neurocomputing
http://dx.doi.org/10.1016/j.neucom.2014.01.063
0925-2312/& 2014 Elsevier B.V All rights reserved.
n Corresponding author.
E-mail address: duybt@vnu.edu.vn (T.D Bui).
Trang 2using maximum operation when forming the weight matrix
are excellent in the presence of erosive noise (1 to 0) while the
models using minimum operation are ideal for dilative noise
(0 to 1)[11] On the other hand, models with maximum operation
cannot recover well patterns with dilative noise and models with
minimum operation cannot recover well patterns with erosive
noise In grey scale or fuzzy valued mode, even though existing
models can recover main parts of the output pattern, noisy parts of
the input pattern affect seriously to the recalled output pattern
Threshold is probably the most effective mechanism so far to deal
with this problem However, the incorrectly recalled parts in the
output are normallyfixed with some pre-calculated value based
on the training input and output pairs Clearly, there are two main
ways to increase noise tolerance capability of the associative
memory models, which are recovering from the noise and
redu-cing the effect of the noise Existing models concentrate on the
first way The work in this paper is motivated from the second
way, which is how to reduce the effect of noisy input patterns to
the recalled output patterns We propose our work based on the
implicative fuzzy associative memories[2], which also belong to
the class of morphological associative memories [11] Instead of
using only rules to store the associations of the input and output
patterns, we also add a certain part of the output patterns
themselves in the weight matrix Depending on the ratio of
association and content of output patterns in the weight matrix,
the effect of noise in the distorted input patterns onto the recalled
output patterns can be reduced Obviously, incorporating the
content of the output patterns would influence the output
selec-tion in the recall phase However, the advantages from the tradeoff
are worth to consider We have conducted experiments in recalling
images from the number dataset and the Corel dataset with both
erosive and dilative noises to confirm the effectiveness of our
model when dealing with noise
The rest of the paper is organized as follows.Section 2presents
background on fuzzy associative memory models We also present
in this section motivational analysis for our work InSection 3, we
describe in detail our model.Section 4 presents analysis on the
properties of the proposed model and experiments to illustrate
these properties
2 Background and motivation
2.1 Fuzzy associative memory models
The objective of associative memories is to recall a predefined
output pattern given the presentation of a predefined input
pattern Mathematically, the associative memory can be defined
as a mapping G such that for a finite number of pairs fðAξ; BξÞ;
ξ¼ 1; …; kg:
The mapping G is considered to have the ability of noise
tolerance if GðA0ξÞ is equal to Bξfor noisy or incomplete version
A0ξ of Aξ The memory is called auto-associative memory if the
pattern pairs are in the form of fðAξ; AξÞ;ξ¼ 1; …; kg The memory
is hetero-associative if the output Bξis different from the input Aξ.
The process of determining G is called learning phase, and the
process of recalling Bξusing G with the presentation of Aξis called
recall phase When G is described by a fuzzy neural network,
and the patterns Aξand Bξare fuzzy sets for everyξ¼1,…,k, the
memory is called fuzzy associative memory (FAM)
The very early FAM models are developed by Kosko in the early
1990s[6], which are usually referred as max–min FAM and
max-product FAM Both of them are single layer feed-forward artificial
neural networks If WA½0; 1mnis the synaptic weight matrix of a
max–min FAM and if AA½0; 1n is the input pattern, then the output pattern BA½0; 1m is computed as follows:
or
Bj¼ ⋁n
i ¼ 1
Wij4Ai ðj ¼ 1…mÞ: ð3Þ Similarly, the max-product FAM produces the output
or
Bj¼ ⋁n
i ¼ 1
Wij:Ai ðj ¼ 1…mÞ: ð5Þ For a set of pattern pairs fðAξ; BξÞ :ξ¼ 1; …; kg, the learning rule used to store the pairs in a max–min FAM, which is called correlation-minimum encoding, is given by the following equa-tion:
or
Wij¼ ⋁k
ξ¼ 1
Bξ
Similarly, the learning rule for the max-product FAM called correlation-product encoding is given by W ¼ B○PAT
Chung and Lee generalized Kosko's model by substituting the max–min or the max-product with a more general max-t product
[7] The resulting model, called generalized FAM (GFAM), can be described in terms of the following relationship between an input pattern A and the corresponding output pattern B:
B ¼ W○TA where W ¼ B○TAT; ð8Þ and the symbol○T denotes the max-C product and C is a t-norm This learning rule is referred as correlation-t encoding
For these learning rules to guarantee the perfect recall of all stored patterns, the patterns A1; …; Ak must constitute an ortho-normal set Fuzzy patterns A; BA½0; 1nare said max-t orthogonal if and only if AT○TB ¼ 0, i.e TðAj; BjÞ ¼ 0 for all j ¼ 1; …; n Conse-quently, A1; …; Ak
is a max-t orthonormal set if and only if the patterns Aξand Aηare max-t orthogonal for everyξaηand Aξis a
normal fuzzy set for everyξ¼ 1; …; k Some research focused on the stability of FAMs and the conditions for perfectly recalling stored patterns are[11–14]
Based on Kosko's max–min FAM, Junbo et al.[9]introduced a new learning rule for FAM which allows for the storage of multiple input pattern pairs The synaptic weight matrix is computed as follows:
where the symbol⊛Mdenotes the min-IMproduct
Liu proposed a model, which is also known as the max–min FAM with threshold [10] The recall phase is described by the following equation:
B ¼ ðW○MðA3cÞÞ3θ: ð10Þ The weight matrix WA½0; 1mnis given in terms of implicative learning and the thresholdsθA½0; 1m and c ¼ ½c1; …; cnTA½0; 1n
are of the following form:
θ¼ ⋀k
ξ¼ 1
Trang 3cj¼ ⋀i A D j⋀ξA LE ijBξ
0 if Di¼ ∅
(
ð12Þ
where LEij¼ fξ: AξjrBξig and Dj¼ fi : LEija∅g
With implicative fuzzy learning, Sussner and Valle established
implicative fuzzy associative memories (IFAMs) [2] IFAMs are
quite similar to the GFAM model of Chung and Lee in the way
that the model is given by a single layer feed-forward artificial
neural network with max-T neurons Here, T is a continuous
t-norm Different from GFAM, the IFAM model includes a bias
term θ¼ ½0; 1n and uses the R-implicative fuzzy learning rule
Given an input pattern AA½0; 1n, the IFAM model produces the
following output pattern BA½0; 1m:
B ¼ ðW○TAÞ3θ; ð13Þ
where
and
θ¼ ⋀k
ξ¼ 1
The fuzzy implication ITis determined by the following equation:
ITðx; yÞ ¼ 3fzA½0; 1 : Tðx; zÞryg 8x; yA½0; 1 ð16Þ
Xiao et al.[15]designed a model that applied the ratio of
input-to-output patterns for the associations:
Wij¼ ⋀k
ξ¼ 1
minðAξ
i; BξjÞ
maxðAξ
Wang and Lu[16] proposed a set of FAM that used division
operator to describe the associations and erosion/dilations for
generalizing the associations
2.2 Motivation
Our study is motivated from the question how to reduce the
effect of noise in distorted patterns onto the recalled output
patterns We base our work on IFAMs [2], particularly with
Lukasiewicz fuzzy conjunction, disjunction and implication [17]
We now present how to incorporate both association and output
content into the weight matrix based on IFAMs
Recall that the Lukasiewicz conjunction is defined as
CLðx; yÞ ¼ 03ðxþy1Þ; ð18Þ
Lukasiewicz disjunction is defined as
DLðx; yÞ ¼ 14ðxþyÞ; ð19Þ
and Lukasiewicz implication is defined as
ILðx; yÞ ¼ 14ðyxþ1Þ: ð20Þ
If AA½0; 1n is the input pattern, and BA½0; 1m is the output
pattern, the learning rule to store the pairs in a Lukasiewicz IFAM
using implication is given by the following equation:
Wij¼ ILðAi; BjÞ ði ¼ 1…n; j ¼ 1…mÞ
The recall phase using conjunction is described by the
follow-ing equation:
Yj¼ CLðWij; AiÞ ðj ¼ 1…mÞ
¼ ⋁n
i ¼ 1
03ðWijþAi1Þ ð22Þ
In order to store both association and output content in the
weight matrix, we modify the learning rule of Lukasiewicz IFAM
using both disjunction and implication as follows:
Wij¼ DLðBj; ILðAi; BjÞÞ ði ¼ 1…n; j ¼ 1…mÞ
¼ 14ðBjþ14ðBjAiþ1ÞÞ
¼ 14ð14ðBjAiþ1ÞþBjÞ
¼ 14ðð1þBjÞ4ð2BjAiþ1ÞÞ
¼ 14ð1þBjÞ4ð2BjAiþ1Þ
¼ 14ð2BjAiþ1Þ ð23Þ And in the recall phase, the conjunction is used with a multi-plication factor of1:
Yj¼1
2 LðWij; AiÞ ðj ¼ 1…mÞ
¼1
2 ⋁n
i ¼ 1
03ðWijþAi1Þ
¼1
2 ⋁n
i ¼ 1
03ð14ð2BjAiþ1ÞþAi1Þ
¼1
2 ⋁n
i ¼ 1
03ðAi42BjÞ
¼1
2 ⋁n
i ¼ 1
Ai42Bj
¼1
2 ⋁n
i ¼ 1
Ai
!
We name our associative memory as association-content asso-ciative memory (ACAM) It can be easy to see that the condition for
W to satisfy the equation WA¼ B is that
Bjr12 ⋁n
i ¼ 1
Ai
!
ð25Þ
If we relax the condition of WijA½0…1, then
Wij¼ 2BjAiþ1; ð26Þ and as a result
Yj¼1
2 ⋁n
i ¼ 1
03ðð2BjAiþ1ÞþAi1Þ
¼1
2 ⋁n
i ¼ 1
032Bj
the equation WA ¼B is satisfied naturally
For a set of pattern pairs fðAξ; BξÞ :ξ¼ 1; …; kg, the weight matrix is constructed with an erosion operator as follows:
Wij¼ ⋀k
ξ¼ 1
2Bξ
3 Generalized association-content associative memory for grey scale patterns
It is clear that we can remove þ1 in the formula of W and 1
in the formula of Y without any effect on our model It can also be seen that our association-content learning rule is similar to morphological associative memory except that there is a multi-plication factor of 2 to Bj This actually represents the portion
of output content to be added to the weight matrix besides the association represented by BjAi More generally, the weight matrix can be constructed as follows:
Wij¼ ⋀k
ξ¼ 1
ðð1ηÞBξjþηðBξjAξiÞÞ
¼ ⋀k
ξ¼ 1
ðBξjηAξ
Trang 4whereηis a factor to control the ratio between the content and
the association to be stored With theηfactor, when the input is
noisy, the noise will have less effect on the recalled patterns
For an input X, the output Y is recalled from W with the
equation:
Yj¼ ⋁m
i ¼ 1
In order to maintain the ability to store an unlimited number
of patterns in the auto-associative case, we keep Wiithe same as in
MAMs[11]:
Wij¼ ⋀k
ξ¼ 1ðBξjAξiÞÞ if i ¼ j
⋀k
ξ¼ 1ðBξjηAξ
iÞ if iaj
8
<
The equation for recalling is then modified as follows:
Yj¼ ⋁
ðηXiþWijÞ3ðXiþWijÞ ð32Þ
Theorem 1 W in Eq.(31)recalls perfectly for all pairs (Aξ, Bξ) if and
only if for eachξ¼ 1; …; k, each column of matrix WξW contains a
zero entry
Proof W recalls perfectly for all pairs (Aξ, Bξ) 3ð⋁i a j
ðηAξ
iþWijÞ3ðAξiþWijÞÞ ¼ Bξj 8 j ¼ 1; …; m
3Bξj ⋁
!
¼ 0 8ξ¼ 1; …; k and 8j ¼ 1; …; m
3Bξjþ ⋀
ðηAξ
iWijÞ4ðAξiWijÞ
!
¼ 0 8ξ¼ 1; …; k and 8j ¼ 1; …; m
3 ⋀
ðBξjηAξ
¼ 0 8ξ¼ 1; …; k and 8j ¼ 1; …; m
3 ⋀n
i ¼ 1
ðWξijWijÞ ¼ 0 8ξ¼ 1; …; k and 8j ¼ 1; …; m ð33Þ
This last set of equations is true if and only if for eachξ¼ 1; …; k
and each integer j ¼ 1; …; m, each row entry of the j-th column of
½WξW contains at least one zero entry □
4 Properties of association-content associative memory
Similar to MAMs[11]and IFAMs[2], our ACAM converges in
one step Moreover, ACAM has unlimited storage capacity, which is
given by the next theorem
Theorem 2 W in Eq (31) recalls perfectly for all pairs (Aξ, Aξ)
(ξ¼ 1; …; k)
Proof Since Wξ
jj¼ AξjAξj ¼ 0 for each j ¼ 1; …; m and all
ξ¼ 1; …; k Hence, for each ξ¼ 1; …; k, each column of ½WξW
contains a zero entry According to Theorem 1, W recalls perfectly
for all pairs (Aξ, Aξ). □
Similar to MAMs and IFAMs, our ACAM can handle erosive
noise effectively with dilation operation in the recalling equation
However, for MAMs and IFAMs, the noise tolerance capability is
good when the number of stored patterns is much smaller than
the length of the input vector, which decides the size of the weight
matrix W This means that MAMs and IFAMs can correct errors
when a large space of storage is waste This reduces the practical
usability of MAMs and IFAMs Our ACAM can compensate the
errors caused by distorted inputs better than MAMs and IFAMs
To compare the effectiveness of our ACAM in handling noise over other well-known associative memories, we have conducted several experiments The six models which we are comparing with are proposed by Junbo et al.[18], Kosko[6], Xiao et al.[15], Sussner and Valle (IFAMs)[2], and Ritter et al (MAMs)[11]
4.1 Experiments with number dataset
This dataset consists offive 5 5 images of numbers from 0 to 4 Using the standard row-scan method, each pattern image is converted into a vector of size 25 With this dataset, the size of the weight matrix
W is 25 25, which is used to store 5 patterns of size 25 With this dataset, we perform experiments on distorted input images with both auto-associative and hetero-associative modes The distorted images contain both erosive and dilative noises (salt and pepper noise) All models are implemented with dilation operator in recalling function where applicable The distorted images can be seen inFig 1 The criterion to evaluate results are the normalized error, which is calculated as follows:
Eð ~B; BÞ ¼J ~B BJJBJ ð34Þ where B is the expected output pattern, ~B is the recovered pattern, andJ J is the L2norm of a vector
Table 1shows the total error of different models when recalling distorted input images in auto-associative mode As can be seen from the table, our ACAM produces the least total error, while Junbo et al.'s model, Xiao et al.'s model, Sussner and Valle's IFAM and Ritter et al.'s MAM produce a similar amount of total error
Fig 1 Auto-associative memory experiments with the number dataset: the first row contains original training images; the second row contains distorted input images; the third, fourth, fifth and sixth row contains output images from Junbo et al.'s model, Xiao et al.'s model, Sussner and Valle's IFAM, and our ACAM respectively.
Table 1 Auto-associative memory experiment result on the number dataset with Junbo
et al.'s model, Kosko's model, Xiao et al.'s model, Sussner and Valle's IFAM, Ritter
et al.'s MAM and our ACAM.
Trang 5Kosko's model produces the most total error This agrees with
what we have mentioned before that Kosko's model cannot even
produce perfect result for perfect input in many cases The reason
other models produce larger total error than our ACAM model is
that these models cannot work well with both erosive and dilative
noises while our ACAM has a mechanism to reduce the effect of
noise This can be seen more clearly inFig 1 In hetero-associative
mode, the pairs of images to remember are images of 0 and 1,
1 and 2, etc.Table 2shows the total error of different models in
this case From the table we can see that our ACAM also produces
the least total error It should be noted that when there is no noise
or only erosive noise, our model performs slightly worse than
IFAMs and MAMs because of the mechanism to reduce the effect of
noise In the presence of only dilative noise, Xiao et al.'s model also
performs better than our ACAM However, this trade-off is worth
to consider because in practice perfect inputs or inputs distorted
by erosive noise only are not common
4.2 Experiments with Corel dataset This dataset includes images selected from the Corel database (Fig 2) The test patterns are generated from input patterns by degrading them with salt and pepper noise, both at 25 percent the number of pixels.Fig 3shows some generated test patterns
In auto-association mode, 10 images are used The result in auto-association mode, which is presented inTable 3, shows our ACAM's effectiveness in handling salt and pepper noise Fig 4
shows samples in which our method visually improves input pattern more than others Hetero-association mode is tested with
10 pairs of images, in which the input image pattern is different from the output image pattern As in the previous test, input patterns are degraded by salt and pepper noise.Table 4also shows how good our ACAM performs compared to other models in the presence of both erosive and dilative noise Fig 5visually com-pares the results of our model to others
Table 2
Hetero-associative memory experiment result on the number dataset with Junbo et al.'s model, Kosko's model, Xiao et al.'s model, Sussner and Valle's IFAM, Ritter et al.'s MAM and our ACAM.
Fig 2 Some images from the dataset used for the experiments.
Fig 3 Test patterns generated from input patterns with salt and pepper noise.
Table 3
Auto-associative memory experiment result on the Corel dataset with Junbo et al.'s model, Kosko's model, Xiao et al.'s model, Sussner and Valle's IFAM, Ritter et al.'s MAM and our ACAM.
Fig 4 Samples from the Corel dataset of which proposed model visually recovers pattern better than other method in auto-associative mode From left to right are patterns salt and pepper noise recovered by Junbo et al.'s model [18] , Kosko's model [6] , Xiao et al.'s model [15] and Sussner and Valle's IFAM [2] , our ACAM model, and the expected result.
Table 4
Hetero-associative memory experiment result on the Corel dataset with Junbo et al.'s model, Kosko's model, Xiao et al.'s model, Sussner and Valle's IFAM, Ritter et al.'s MAM and our ACAM.
Trang 65 Conclusion
In this paper, we proposed a new FAM that captures both content
and association of patterns While still possessing vital advantages of
existing FAMs, our model has better noise tolerance when both
erosive and dilative noises are present This is achieved by sacrificing
the reduction of performance in special cases (no noise, only erosive
or dilative noise) We have conducted experiments on different data
sets to prove the efficiency of the proposed FAM The obtained
results hint that the improvement in capturing both pattern content
and associations can be effective
It is noted that the paper is only thefirst step to show a way of
reducing the effect of noise in FAMs There are many ways in
which the paper can be extended First of all, mathematical
analysis of how the effect of noise is reduced is an interesting
problem to solve Secondly, besides combining content of output
with association based on IFAMs and MAMs, using this approach
with other existing FAMs would be a nice try Finally, it is worth
to compare to and integrate with other associative memories
besides FAMs, such as associative memories based on
discrete-time recurrent neural networks[19–21]
Acknowledgements
This work is supported by Nafosted Research Project no
102.02-2011.13
References
[1] J.J Hopfield, Neural networks and physical systems with emergent collective
computational abilities, Proc Natl Acad Sci U S A 79 (1982) 2554–2558
[2] P Sussner, M.E Valle, Implicative fuzzy associative memories, IEEE Trans.
Fuzzy Syst 14 (6) (2006) 793–807
[3] M.-J Kim, I Han, K.C Lee, Fuzzy associative memory-driven approach to
knowledge integration, in: 1999 IEEE International Fuzzy Systems Conference
Proceedings, 1999, pp 298–303.
[4] S Shahir, X Chen, Adaptive fuzzy associative memory for on-line quality
control, in: Proceedings of the 35th South-Eastern Symposium on System
Theory, 2003, pp 357–361.
[5] Z Wang, J Zhang, Detecting pedestrian abnormal behavior based on fuzzy
associative memory, in: Fourth International Conference on Natural
Computa-tion, 2008, pp 143–147 〈 http://dx.doi.org/10.1109/ICNC.2008.396 〉.
[6] B Kosko, Neural Networks and Fuzzy Systems: A Dynamical Systems
Approach to Machine Intelligence, Prentice Hall, Englewood Cliffs, NJ, 1992
[7] F Chung, T Lee, On fuzzy associative memories with multiple-rule storage
capacity, IEEE Trans Fuzzy Syst 4 (3) (1996) 375–384
[8] A Blanco, M Delgado, I Requena, Identification of fuzzy relational equations
by fuzzy neural networks, Fuzzy Sets Syst 71 (1995) 215–226
[9] F Junbo, J Fan, S Yan, A learning rule for FAM, in: 1994 IEEE International
Conference on Neural Networks, 1994, pp 4273–4277.
[10] P Liu, The fuzzy associative memory of max–min fuzzy neural networks with
threshold, Fuzzy Sets Syst 107 (1999) 147–157
[11] G Ritter, P Sussner, J.D de Leon, Morphological associative memories, IEEE
Trans Neural Netw 9 (1998) 281–293
[12] Z Zhang, W Zhou, D Yang, Global exponential stability of fuzzy logical BAM
neural networks with Markovian jumping parameters, in: 2011 Seventh
International Conference on Natural Computation, 2011, pp 411–415.
[13] S Zeng, W Xu, J Yang, Research on properties of max-product fuzzy
associative memory networks, in: Eighth International Conference on
Intelli-gent Systems Design and Applications, 2008, pp 438–443.
[14] Q Cheng, Z.-T Fan, The stability problem for fuzzy bidirectional associative memories, Fuzzy Sets Syst 132 (1) (2002) 83–90
[15] P Xiao, F Yang, Y Yu, Max–min encoding learning algorithm for fuzzy max-multiplication associative memory networks, in: 1997 IEEE International Conference on Systems, Man, and Cybernetics, 1997, pp 3674–3679 [16] S.T Wang, H.J Lu, On new fuzzy morphological associative memories, IEEE Trans Fuzzy Syst 12 (3) (2004) 316–323
[17] W Pedrycz, F Gomide, An Introduction to Fuzzy Sets: Analysis and Design, Complex Adaptive Systems, MIT Press, 1998
[18] F Junbo, J Fan, S Yan, An encoding rule of FAM, in: Singapore ICCS/ISITA '92,
1992, pp 1415–1418.
[19] Z Zeng, J Wang, Analysis and design of associative memories based on recurrent neural networks with linear saturation activation functions and time-varying delays, Neural Comput 19 (8) (2007) 2149–2182
[20] Z Zeng, J Wang, Design and analysis of high-capacity associative memories based on a class of discrete-time recurrent neural networks, IEEE Trans Syst Man Cybern Part B: Cybern 38 (6) (2008) 1525–1536
[21] Z Zeng, J Wang, Associative memories based on continuous-time cellular neural networks designed using space-invariant cloning templates, Neural Netw 22 (2009) 651–657
The Duy Bui got his Bachelor degree in Computer Science from University of Wollongong, Australia in
2000 and his Ph.D in Computer Science from Univer-sity of Twente, the Netherlands in 2004 He is now working at Human–Machine Interaction Laboratory, University of Engineering and Technology, Vietnam National University, Hanoi His research includes Machine Learning, Human–Computer Interaction, Computer Graphics and Image Processing.
Thi Hoa Nong received the Master of Science in Information Technology from Thainguyen University
in 2006 She is pursuing Ph.D degree in Computer Science at University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam She is now a lecturer in Thainguyen University, Vietnam Her current research interests include artificial intelligence, machine learning.
Trung Kien Dang got M.Sc in Telematics from University
of Twente, the Netherlands, in 2003 and received Ph.D.
in Computer Science from University of Amsterdam, the Netherlands in 2013 His research includes machine learning, 3D model reconstruction and video log analysis Fig 5 From left to right are patterns from the Corel dataset recovered from salt and pepper noise in hetero-associative mode by Junbo et al.'s model [18] , Kosko's model [6] , Xiao et al.'s model [15] and Sussner and Valle's IFAM [2] , our ACAM model, and the expected result.