=Paper=
{{Paper
|id=Vol-1624/paper24
|storemode=property
|title=Data Retrieval and Noise Reduction by Fuzzy Associative Memories
|pdfUrl=https://ceur-ws.org/Vol-1624/paper24.pdf
|volume=Vol-1624
|authors=Irina Perfilieva,Marek Vajgl
|dblpUrl=https://dblp.org/rec/conf/cla/PerfilievaV16
}}
==Data Retrieval and Noise Reduction by Fuzzy Associative Memories==
https://ceur-ws.org/Vol-1624/paper24.pdf
Data Retrieval and Noise Reduction by Fuzzy
Associative Memories
Irina Perfilieva, Marek Vajgl
University of Ostrava,
Institute for Research and Applications of Fuzzy Modeling,
Centre of Excellence IT4Innovations,
30. dubna 22, 701 03 Ostrava 1,
Czech Republic
E-mails: Irina.Perfilieva,Marek.Vajgl@osu.cz
Abstract. A novel theoretical background of fuzzy associative memo-
ries (FAM) is proposed. A framework of formal concept analysis is used
for a new working theory of FAM. Two principal activities of FAM are
formalized : data retrieval and noise reduction. It is shown that the prob-
lem of data retrieval is connected with solvability and eigen sets of a
certain system of fuzzy relation equations. The differentiation of FAM
models according to their ability to reduce noise is defined. It is shown
how the choice of formal context determines a type of noise that can be
reduced by the corresponding retrieval mechanism. Finally, we propose
a fast algorithm of data retrieval.
1 Introduction
One of the first publications devoted to fuzzy associative memories (FAM) has
been made by Kosko - [6]. The FAM has been characterized as a single-layer
feedforward neural net performing a nonlinear matrix-vector multiplication. This
approach was later extended with the purpose to increase the storage capacity
(e.g. [4]). Significant progress was achieved by introduction of learning impli-
cation rules [3, 5], that afterwards led to implicative fuzzy associative memory
(IFAM) with implicative fuzzy learning. A justification of validity of a certain
IFAM model was discussed in [14] where the characterization of one type of
suppressed noise - eroded - was proposed.
In the current contribution, we use the framework of formal concept analysis
[17] and propose a working theory of FAM. Let us remark that the language
and technique of formal concept analysis is used in other theories, e.g., fuzzy
property-oriented concept lattices, as well as in applications such as modeling
and processing of incomplete knowledge in information systems [1, 2].
We formalize two principal activities of FAM: data retrieval and noise re-
duction. We use the proposed formalism and show that the problem of data
retrieval is connected with solvability and eigen sets of a certain system of fuzzy
relation equations. We differentiate FAM models according to their ability to
reduce noise and show how the choice of formal context determines a type of
noise that can be reduced by the corresponding retrieval mechanism.
From the technical point of view, we extend the theory of FAM by using
a general algebraic structure instead of a specific one (Lukasiewicz algebra in
[14]) and by enlarging the set of autoregressive fuzzy associative memory models
(AFAM). We propose a formal characterization of AFAM models and show the
way of various modifications. We analyze the retrieval mechanism of AFAM and
its ability to remove noise. We show that the larger is the amount of noise, the
greater should be the fuzzy relation that models retrieval with noise reduction.
Further, we show how the type of removable noise depends on which type of
AFAM models is applied. Finally, we construct a fast algorithm of data retrieval
and give illustration of the noise reduction.
2 Preliminaries
2.1 Implicative fuzzy associative memory
In this Section, we discuss underlying assumptions related to autoregressive fuzzy
associative memories (AFAM) using denotation in [14]. We formalize the retrieval
mechanism and the problem of noise reduction.
Let us propose the following formalization of AFAM. A database
D = {x1 , . . . , xp },
of objects (images, patterns, signals, texts, etc.) is represented by normal fuzzy
sets such that every xk , k = 1, . . . , p, is a map xk : X → [0, 1], where X =
{u1 , . . . , un } is a universe. The problem is to find a model of D together with a
retrieval mechanism such that every element xk ∈ D can be successfully retrieved
even from its noisy version.
According to [14], a model of AFAM for database D can be identified with the
3-tuple (W, θ, ), consisting of a fuzzy relation W : X × X → [0, 1], a bias vector
θ ∈ [0, 1]n , and a set-relation composition : [0, 1]n×n ×[0, 1]n ×[0, 1]n → [0, 1]n ,
such that for all xk ∈ D,
xk = W xk ∨ θ, k = 1, . . . , p. (1)
We say that (1) represents a retrieval mechanism in AFAM and that this mech-
anism reduces noise, if
xk = W ek ∨ θ, k = 1, . . . , p,
x (2)
where xek is a noisy version of xk .
In [14], an implicative model of AFAM has been proposed. The model uses
Lukasiewicz algebra of operations on [0, 1], so that fuzzy relation W is expressed
in the implicative form
p
^
W (ui , uj ) = (xk (ui ) → xk (uj ), (3)
k=1
and other constituents are as follows:
p
^
θi = xk (ui ), i = 1, . . . , n,
k=1
is the sup −⊗-composition.
For a given input x, AFAM returns the output in accordance with (1) and
the assignment given above, so that
n
_
W x∨θ = (W (ui , uj ) ⊗ x(ui )) ∨ θ.
i=1
The proposed in [14] AFAM is able to reduce an eroded noise, i.e. if x is less
than some database element, say xk , then the retrieved output is close to xk as
well.
2.2 Algebraic background
In this Section, we step aside from the terminology of associative memories and
introduce the algebraic background of what will be proposed as a new model of
AFAM.
Let L = hL, ∨, ∧, ∗, →, 0, 1i be a fixed, complete, integral, residuated, com-
mutative l-monoid (a complete residuated lattice). We remind the main charac-
teristics of this structure: hL, ∨, ∧, 0, 1i is a complete bounded lattice, hL, ∗, →, 1i
is a residuated, commutative monoid.
Let X be a non-empty set, LX a class of (L-valued) fuzzy sets on X and
X×X
L a class of (L-valued) fuzzy relations on X. Fuzzy sets and fuzzy relations
are identified with their membership functions, i.e. elements from LX and LX×X ,
respectively. A fuzzy set A is normal if there exists xA ∈ X such that A(xA ) = 1.
The (ordinary) set Core(A) = {x ∈ X | A(x) = 1} is the core of the normal
fuzzy set A. Fuzzy sets A ∈ LX and B ∈ LX are equal (A = B), if for all x ∈ X,
A(x) = B(x). A fuzzy set A ∈ LX is less than or equal to a fuzzy set B ∈ LX
(A ≤ B), if for all x ∈ X, A(x) ≤ B(x).
The lattice operations ∨ and ∧ induce the union and intersection of fuzzy
sets, respectively. The binary operations ∗ and → of L are used for set-relation
compositions of the types sup −∗ or inf − → that are usually denoted by ◦ and
. where _
(R ◦ A)(y) = (R(x, y) ∗ A(x)),
x∈X
and ^
(R . A)(y) = (R(x, y) → A(x)).
x∈X
We say that compositions ◦ and . are skew adjoint, which means that for every
A, B ∈ LX , R ∈ LX×X , the following holds:
R ◦ A ≤ B ↔ A ≤ Rop . B,
where Rop (x, y) = R(y, x).
Let us remind that the ◦ composition was introduced by L. Zadeh [18] in the
form max − min.
3 Fuzzy Preorders and Their Eigen Sets
In this Section, we recall basic facts about fuzzy preorder relations. Then we
characterize eigen sets of fuzzy preorder relations.
Our interest to fuzzy preorder relations is connected with the analysis of the
expression in (3) – a representation of the AFAM model. This is the representa-
tion of the so called Valverde (fuzzy) preorder [16].
3.1 Fuzzy preorders
A binary fuzzy relation on X is a ∗-fuzzy preorder of X, if it is reflexive and
∗-transitive. The fuzzy preorder Q∗ ∈ LX×X , where
^
Q∗ (x, y) = (Ai (x) → Ai (y)), (4)
i∈I
is generated by an arbitrary family of fuzzy sets (Ai )i∈I of X.
Remark 1. The fuzzy preorder Q∗ is often called Valverde order determined by
the family of fuzzy sets (Ai )i∈I of X (see [16] for details).
If Q is a fuzzy preorder on X, then Qop is a fuzzy preorder on X as well.
3.2 Eigen sets of fuzzy preorders
In this Section, we show that fuzzy preorder Q∗ (given by (4)) generated by
fuzzy sets (Ai )i∈I ⊆ LX , is the greatest solution to the system of fuzzy relation
equations
W ◦ Ai = Ai , i ∈ I, (5)
where W denotes an unknown fuzzy relation. At the same time, (Q∗ )op is the
greatest solution to the system of fuzzy relation equations
W . Ai = Ai , i ∈ I. (6)
Moreover, we show that there exists a binary preorder that gives a solution to
(5) and (after the “transposition”) to (6).
Proposition 1. Let (Ai )i∈I ⊆ LX , be a family of fuzzy sets of X and a fuzzy
preorder Q∗ be generated by this family in the sense of (4). Then Q∗ is the
greatest solution to the system of fuzzy relation equations (5) and (Q∗ )op is the
greatest solution to the system of fuzzy relation equations (6).
Proof. It is obvious that both systems are solvable - this is because the identity
(fuzzy) relation is a solution to (5) and (6). This fact implies that fuzzy relation
Q∗ is the greatest solution to the system (5). Let us prove the second claim.
The following chain of equivalences can be easily obtained from the first
claim:
!
_
(∀y) (Q∗ (x, y) ∗ Ai (x)) ≤ Ai (y) ⇔ (∀y)(∀x)(Q∗ (x, y)∗Ai (x) ≤ Ai (y)) ⇔
x∈X
^
(∀x)(∀y)(Ai (x) ≤ Q∗ (x, y) → Ai (y)) ⇔ (∀x) Ai (x) ≤ (Q∗ (x, y) → Ai (y)) .
y∈X
By reflexivity of Q∗ ,
^
(Q∗ (x, y) → Ai (y)) ≤ Ai (x).
y∈X
Therefore,
(Q∗ )op . Ai = Ai .
Corollary 1. Let the assumptions of Proposition 1 be fulfilled. Then fuzzy sets
Ai , i ∈ I, are eigen fuzzy sets of the relation Q∗ ((Q∗ )op ) with respect to com-
position ◦ (.).
By Proposition 1, fuzzy relation Q∗ ((Q∗ )op ) is the greatest solution of the
system (5) (similarly, fuzzy relation (Q∗ )op is the greatest solution of the system
(6)). Let us show that there are smaller fuzzy relations that solve the system (5)
(resp. the system (6)). Moreover, these smaller relations are ordinary (binary)
preorders on X.
Proposition 2. Let (Ai )i∈I ⊆ LX be a family of fuzzy sets of X and fuzzy
preorder Q∗ be generated by this family in the sense of (4). Let ∆ : L → L be
the following unary operation on L:
(
1, if a = 1,
∆(a) =
0, otherwise.
Then ∆(Q∗ ) is a solution to the system (5) and (∆(Q∗ ))op is a solution to the
system (6). Moreover, ∆(Q∗ ) is an ordinary (binary) preorder on X.
Proof. At first, we prove that ∆(Q∗ ) is a solution to (5). This fact easily follows
from the following two inequalities:
∆(Q∗ ) ◦ Ai ≤ Q∗ ◦ Ai = Ai , i ∈ I,
∆(Q∗ ) ◦ Ai ≥ Ai .
The first inequality is due to ∆(Q∗ ) ≤ Q∗ . The second inequality follows from
reflexivity of ∆(Q∗ ).
At second, we show that ∆(Q∗ ) is a preorder relation on X. The property of
reflexivity is inherited from Q∗ .To prove transitivity we choose t, u, v ∈ X and
consider the case ∆(Q∗ )(t, u) = 1 and ∆(Q∗ )(u, v) = 1. It is easy to see that
∆(Q∗ )(t, u) ∗ ∆(Q∗ )(u, v) = Q∗ (t, u) ∗ Q∗ (u, v) ≤ Q∗ (t, v).
We refer to ∆(Q∗ ) as to a binary “skeleton” of Q∗ .
4 New AFAM Models
In this Section, we put a bridge between the theory, presented in Section 3,
and the theory of autoregressive fuzzy associative memories (AFAM), presented
in Section 2. We propose a new concept of adjoint AFAM models that share
a common fuzzy preorder relation. We characterize types of noise that can be
reduced by retrieval in corresponding AFAM models.
4.1 Adjoined AFAM Models
Let us choose and fix complete residuated lattice with the support L = [0, 1] and
database D = {x1 , . . . , xp } of 2D [0, 1]-valued (gray scaled) images. We assume
that the images in D are represented by n-dimensional vectors, so that each
vector is a sequence of image rows. We identify every image with a fuzzy set on
n̄ = {1, 2, . . . , n}, so that xk ∈ [0, 1]n , k = 1, . . . , p. We additionally assume that
all fuzzy sets in D are normal.
Definition 1. We say that a pair (W, ), where W ∈ [0, 1]n×n is a fuzzy relation
and : [0, 1]n×n × [0, 1]n → [0, 1]n is a set-relation composition, is an AFAM
model of database D, if for all xk ∈ D,
xk = W xk , k = 1, . . . , p. (7)
We say that two AFAM models (W, ) and (W, e ) of D are adjoint, if there
exists a complete residuated lattice L such that is of the sup −∗ type, e is of
the inf − → type and and e are skew-adjoint.
By (7) and the terminology of AFAM, any element from D is successfully
retrieved from its sample in a corresponding AFAM model. On the other hand,
according to the terminology of set-relation compositions, the same equation
(7) characterizes any element from D as an eigen set of the fuzzy relation with
respect to a certain composition - both are constituents of the corresponding
AFAM model.
Example 1. Let us choose a complete residuated lattice L on [0, 1] and give
two examples of adjoint AFAM models. Following (4), we construct the fuzzy
preorder relation Q∗ , such that
p
^
Q∗ (i, j) = (xk (i) → xk (j)), (8)
k=1
and take its binary skeleton ∆(Q∗ ). By Proposition 1, the two pairs (Q∗ , ◦) and
((Q∗ )op , .) are examples of adjoint AFAM models. By Proposition 2, the two
pairs (∆(Q∗ ), ◦) and (∆(Q∗ )op , .) are examples of adjoint AFAM models too.
4.2 AFAM and Noise Reduction
Let us characterize types of noise that can be removed/reduced by the retrieval
mechanisms of adjoint AFAM models.
Definition 2. Let (W, ) be a model of AFAM with respect to database D, x̃ ∈
[0, 1]n , and x̃ 6∈ D. We say that x̃ is a noisy version of some element x ∈ D,
that can be removed by the retrieval in (W, ), if
x=W x̃. (9)
Proposition 3. Let D be a database, L a complete residuated lattice on [0, 1],
and (W, ◦), (W, .) adjoint AFAM models. Let moreover, fuzzy relation W be
reflexive. Then
(i) if (W, ◦) removes a noisy version x̃ of the element x ∈ D, then x̃ is an
eroded version of x, i.e. x̃ ≤ x;
(ii) if (W, .) removes a noisy version x̃ of the element x ∈ D, then x̃ is a
dilated version of x, i.e. x̃ ≥ x.
Proof. We give the proof of the case (i). Assume that x = (x1 , . . . xn ), x̃ =
(x̃1 , . . . x̃n ) and W ◦ x̃ = x. Then for any j = 1, . . . , n,
n
_
xj = (W (i, j) ∗ x̃i ) ≥ W (j, j) ∗ x̃j ) = x̃j .
i=1
Therefore, x̃ ≤ x.
5 AFAM and Formal Concept Analysis
In this section, we explain the relationship between the proposed theory of AFAM
and the theory of formal concept analysis (FCA) [17]. For this purpose, we adapt
the terminology of FCA to the proposed above analysis of fuzzy associative
memories.
We choose and fix a finite set X and a complete residuated lattice L. The
following formal context K = (A, R, I ) where A ⊆ LX is a set of objects,
R ⊆ LX×X is a set of attributes, and I is an incidence relation on A × R,
is proposed. We say that an object (dataset) A possesses an attribute R, if
the latter is an AFAM model of A via the composition . Equivalently and in
accordance with Definition 1, I (A, R) = 1 if and only if A ∈ A is an eigen
fuzzy set of R ∈ R with respect to the composition .
Let us choose and fix a formal context K with the given above specification.
A couple (A, W) is a formal concept of K, if A ⊆ A is a dataset, W ⊆ R is a
set of fuzzy relations such that A ∈ A if and only if A is an eigen fuzzy set of
any W ∈ W with respect to the composition , and vise versa, W ∈ W if and
only if (W, ) is an AFAM model of A.
Using both languages, we say that every element in dataset A has every
attribute in W, i.e. can be successfully retrieved from its sample using any AFAM
model (W, ) where W ∈ W and composition is clear from the context K.
Below, we analyze, how the problems of data retrieval and noise reduction
can be formalized in terms of context analysis. We formulate two assertions
and characterize formal concepts where datasets as formal concept objects are
connected with retrieval models as formal concept attributes. The proofs of both
below given statements can be obtained from Propositions 1,2.
Proposition 4. Let X be a set, L a complete residuated lattice, K = (LX , LX×X ,
I◦ ) a formal context and A ⊆ LX a dataset of objects. Then the smallest concept
with objects from A includes as attributes all fuzzy relations R ∈ LX×X such
that ∆(Q∗ ) ≤ R ≤ Q∗ where Q∗ and ∆(Q∗ ) are specified in Propositions 1,2.
Proposition 5. Let X be a set, L a complete residuated lattice, K = (LX , LX×X ,
I. ) be a formal context and A ⊆ LX a dataset of elements. Then the smallest
concept with elements from A includes as attributes all fuzzy relations R ∈ LX×X
such that ∆(Q∗ )op ≤ R ≤ (Q∗ )op where (Q∗ )op and ∆(Q∗ )op are specified in
Propositions 1,2.
The difference between Propositions 4 and 5 is in the choice of formal context.
The latter determines a type of composition that connects an element from a
dataset with the corresponding model.
By Definition 2, a noisy and ideal elements from A are connected by equation
(9). The following relationship is an easy consequence of (9): the larger is an
amount of noise, the greater should be a fuzzy relation that models retrieval
with noise reduction and vise versa. This fact is illustrated in Fig. 2 where
we demonstrate two results of noise reduction: the one is based on the model
(Q∗ , ◦) and the other one - on the model (∆(Q∗ ), ◦). It easily observed that
model (Q∗ , ◦) reduces a larger amount of noise than the other model (∆(Q∗ ), ◦).
This is because fuzzy relation Q∗ is greater than fuzzy relation ∆(Q∗ ).
Let us construct a formal concept of K, suitable for characterization of an
AFAM model with the ability of noise reduction. For this purpose we differentiate
AFAM models according to “degrees of fuzziness” of their fuzzy relations. The
latter will be defined as X
δ(W ) = W (x, y),
x,y∈X
where W ∈ L X×X
. It is easy to see that for two fuzzy relations Q∗ (given by (4))
and its binary skeleton ∆(Q∗ ), the following inequality holds: δ(∆(Q∗ )) ≤ δ(Q∗ ).
A formal concept (A, WD ) of K with the ability of noise reduction is a couple
(A, WD ), where A ∈ A, if and only if A is an eigen fuzzy set of any W ∈ WD
such that δ(W ) ≥ D, and vise versa, W ∈ WD , if and only if δ(W ) ≥ D and
(W, ) is an AFAM model of A. The value of D regulates the amount of noise
and should be less or equal than δ(Q∗ ).
6 Illustration
The aim of this Section is to give illustrations to the theoretical results of this
paper. We use gray scaled images with the range [0, 1], where 0 (1) represents
the black (white) color. We choose two different datasets of images, both were
artificially created from open access databases. These sets contain 2D images of
40 × 30 and 120 × 90 pixels, respectively. All images are represented by corre-
sponding fuzzy sets with values in [0, 1].
6.1 Algorithms of data retrieval
We tested the two AFAM models (Q∗ , ◦) and (∆(Q∗ ), ◦), given in Example 1. The
first experiment is to verify data retrieval. Both models successfully passed the
verification. For the model that is based on the relation ∆(Q∗ ), we elaborated a
fast algorithm of data retrieval that uses the fact that ∆(Q∗ ) is actually a binary
relation.
We compare average execution time of the two corresponding algorithms and
conclude that the algorithm based on ∆(Q∗ ) is up to thirty time faster than that
based on Q∗ , see Table 1.
Image size Run-time 1 Run-time 2
40 x 30 5.40 0.195
120 x 90 435.80 13.79
Table 1. Run-time (in seconds) of the two algorithms of data retrieval that are based
on models (Q∗ , ◦) (left column) and (∆(Q∗ ), ◦) (right column).
6.2 Reduction of Noise
The second experiment is to analyze the influence of different retrieval AFAM
models (Q∗ , ◦) and (∆(Q∗ ), ◦) on eroded noise. For this purpose, we added 70%
pepper noise to the original image in Fig. 1. The algorithm of data retrieval,
based on the model (Q∗ , ◦), was more efficient than that, based on the model
(∆(Q∗ ), ◦), see Fig. 2.
7 Conclusion
A new theory of autoregressive fuzzy associative memories (AFAM) has been
proposed. It extends [14] by using general algebraic structures and new types of
autoregressive fuzzy associative memory models. We showed how the proposed
theory is connected with systems of fuzzy relation equations and eigen sets of
their solutions. We proposed a new concept of adjoint AFAM models that share
Fig. 1. Original image without noise (left) and with 70 % pepper noise (right).
Fig. 2. Noise reduction based on the model (Q∗ , ◦) (left) and on the model (∆(Q∗ ), ◦)
(right).
a common fuzzy preorder relation. We characterized two types of noise that
can be reduced by retrieval in corresponding AFAM models. Two problems have
been discussed: data retrieval and noise reduction.
The relationship between the AFAM data retrieval and the formal concept
analysis has been analyzed. We proposed a new working theory of FAM and
formalized it in the language of formal concept analysis. We characterized two
principal activities of FAM: data retrieval and noise reduction. We used the
proposed formalism and showed that the problem of data retrieval is connected
with solvability and eigen sets of a certain system of fuzzy relation equations.
We differentiated FAM models according to their ability to reduce noise and
showed how the choice of formal context determines a type of noise that can be
reduced by the corresponding retrieval mechanism.
Finally, we proposed a fast algorithm of data retrieval that is based on an
AFAM model with a binary fuzzy preorder.
Acknowledgment This paper was supported by the NPUII project IT4I XS
with the number LQ1602.
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