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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 1

Quantum-Inspired Multidirectional AssociativeMemory With a Self-Convergent Iterative Learning

Naoki Masuyama, Member, IEEE, Chu Kiong Loo, Senior Member, IEEE,Manjeevan Seera, Senior Member, IEEE, and Naoyuki Kubota, Member, IEEE

Abstract— Quantum-inspired computing is an emergingresearch area, which has significantly improved the capabilitiesof conventional algorithms. In general, quantum-inspired hop-field associative memory (QHAM) has demonstrated quantuminformation processing in neural structures. This has resultedin an exponential increase in storage capacity while explainingthe extensive memory, and it has the potential to illustratethe dynamics of neurons in the human brain when viewedfrom quantum mechanics perspective although the application ofQHAM is limited as an autoassociation. We introduce a quantum-inspired multidirectional associative memory (QMAM) with aone-shot learning model, and QMAM with a self-convergentiterative learning model (IQMAM) based on QHAM in this paper.The self-convergent iterative learning enables the network to pro-gressively develop a resonance state, from inputs to outputs. Thesimulation experiments demonstrate the advantages of QMAMand IQMAM, especially the stability to recall reliability.

Index Terms— Fuzzy inference, multidirectional associativememory, neural network, quantum-inspired computing (QIC).

I. INTRODUCTION

REPLICATION of the outstanding functions of the humanbrain in a computer, based on analysis and modeling

of the essential functions of a biological neuron and itscomplicated networks has recently become an active researchfield. Several studies in this field have revealed successfuldevelopments of learning and memory which inspired byneural architectures in the brain. The unification of the conceptof quantum information and computer science is regarded asone of the emerging approaches to bring new understandingsto this field, which is called as quantum-inspired comput-ing (QIC) [1]–[3]. In general, from the engineering view,quantum mechanics (QM) has been developed as a theoryto explain the fundamental principles of substance. QM pro-vides several mathematical concepts, such as duality of wavesand particles, complementarity, and nonlocality, to improvethe comprehension of the microworld. From the biological

Manuscript received February 2, 2016; revised June 11, 2016 andJanuary 4, 2017; accepted January 7, 2017. This work was supportedby the UM Grand Challenge from the University of Malaya underGrant GC003A-14HTM.

N. Masuyama and C. K. Loo are with the Faculty of Computer Scienceand Information Technology, University of Malaya, 50603 Kuala Lumpur,Malaysia (e-mail: [email protected]; [email protected]).

M. Seera is with the Faculty of Engineering, Computing and Science,Swinburne University of Technology Sarawak Campus, 93350 Kuching,Malaysia (e-mail: [email protected]).

N. Kubota is with the Graduate School of System Design, Tokyo Metropol-itan University, Tokyo 191-0065, Japan (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNNLS.2017.2653114

perspective, on the other hand, it is hypothesized that QM isbased on mesoscopic features in the physical and biological orphysiological processes of the brain [4], and it has the potentialto illustrate the dynamics of neurons in the human brain bythe quantum information. In fact, in the internal structure ofneuron in the brain, the presence of the two quantum states intubulin, which are proteins of the size 4 nm×8 nm and havinga 20-nm gap between the synapse, suggest that artificial neuralnetworks would be handled as a descriptive subject from QMperspective. In other words, the QIC is expected to be possibleto bring a new standpoint for cognitive process of brain froma biological viewpoint [5].

Conventionally, several types of artificial neural networkswith concept of QM have been introduced and their potentialwith superior abilities to solve problems pertaining recog-nition, classification, and optimization shown [6]–[10]. Inregard as an artificial associative memory model, Rigatosand Tzafestas [11] introduced a quantum-inspired hopfieldassociative memory (QHAM), and showed its superior ability.This model applies a concept of fuzzy inference to weightmatrix satisfying the features of QM, referred to as parallelismand unitarity, in the eigenvector spaces of decomposed weightmatrices. Although the model has the certain advantages,QHAM is only able to perform the autoassociation due to thesingle-layer network, and the learning algorithm of QHAM isnot incremental.

In general, the human thought is always associative, whichis able to perform not only the autoassociation, but also theheteroassociation. We are able to acquire new informationwithout collapsing the memorized information. Reviewing theQHAM from properties of associative memory, the QHAMis insufficient to represent the associative memory in humanbrain due to its network architecture and learning algorithm.We attempt to resolve above-mentioned issues in this paperby structural and algorithmic perspectives, while maintainingquantum information processing based on the QHAM. Thecontributions of this paper are as follows.

1) Extending an autoassociation model as QHAM to amultidirectonal model called quantum-inspired multidi-rectional associative memory (QMAM), which is able torealize the heteroassociation, and collaterally contributeto the stability of recall reliability.

2) Quantum information processing can be observed inthe heteroassociative neural architecture by applyingthe fuzzy inference to represent the synaptic connec-tions in the decomposed weight matrices. A correspon-dence between QM and fuzzy inference is demonstrated

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2 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

through the similarity within the descriptions of the neu-ronal state and quantum state based on the Schrödinger’sdiffusion equation.

3) The mathematical proofs show that the weight matricesin heteroassociation model satisfies the features of QM:1) the existence of superposition in eigenvector spacesof decomposed weight matrices, and 2) the transition ofeigenvector spaces are performed by unitary operator.Thus, the association process of proposed models canbe considered as the result of a quantum measurement,which is a nonlinear mapping that can be represented interms of the eigenvector spaces of decomposed weightmatrices.

4) The self-convergent iterative incremental learning algo-rithm with a nonlinear output function are introducedto QMAM, which is specified as QMAM with a self-convergent iterative learning (IQMAM). This iterativelearning algorithm is able to progressively develop a res-onance state in the weight matrices without overlearning,and its online learning process explicitly incorporates thedesired information with maintaining the properties ofquantum information processing.

Incidentally, the discussion of artificial neural networksin QIC is based on the similarity to the concept of QM,specifically, the linear similarity within the descriptions of theneuronal and quantum states. Thus, in the current situation, itshould be noted that the contents of this paper are not derivedfrom the QM in the strict physics standpoint.

This paper is divided as follows. A literature review onimprovements for artificial associative models is first given inSection II. Section III describes the fundamental concept ofquantum-inspired model, and mathematical proofs for paral-lelism and unitarity in the proposed model. Fundamentals ofthe proposed models are presented in Section IV. Section Vpresents simulation experiments to evaluate the proposed mod-els. Concluding remarks are presented in Section VI.

II. LITERATURE REVIEW

HAM is regarded as a fundamental model of associativememory [12]. One of the limitations of the HAM is thatit performs only an autoassociation. To tackle this issue,Kosko developed a bidirectional associative memory [13], andHagiwara further developed it to a MAM [14]. In general,these fundamental associative memory models apply an off-line, an one-shot learning rules based on the Hebb learning,thus the low noise tolerance and low memory capacity areunavoidable issues.

A number of algorithms are introduced in improving theabilities of the associative memory model. The combina-tion of Hebb learning and pseudo relaxation learning, calledQuick-learning algorithm [15], performs fast learning whilemaintaining superior memory capacity and noise tolerance.Zheng et al. [16] introduced the asymmetric connections tothe associative memory as a structural improvement. Thegiven patterns can be assigned as locally asymptotically stableequilibria of the network by training a single-layer feed-forward network. Valle [17] introduced a projection matrixbased learning, which is an effective approach although the

learning process is not incremental. Several studies focus onthe time delay in networks to reduce spurious equilibriumunder random initial states and generalize the design of autoas-sociative memory [18], [19]. In other approaches, a high-dimensional neuron is utilized in obtaining higher capabilityof associative memory and network stability in the associationprocess [20]–[22]. Although the performance of these modelsis improved, the dynamics of learning and recall processeshave the tendency to be greatly complicated.

Based on human brain modeling and physiological exper-iments, various types of biological inspired approaches havebeen introduced, such as the effectiveness of sparseness inthe human brain [23] and the chaotic behavior in the neuronsof a human brain. It is regarded that the memory and recallfunctions of the human brain are affected by chaotic behavior.Conventionally, several studies of associative memory modelswith chaotic neuron have been introduced to imitate the humanbrain behavior [24], [25]. Although these chaotic associativememory models perform with superior abilities in terms ofmemory capacity, resistance for noise and avoidance of spu-rious attractors, the network complexity will increase.

An emerging research is applying the knowledge of QMto artificial neural networks [8]–[10]. In terms of associativememory model, Rigatos and Tzafestas [11] introduced simpleand effective model, called QHAM. It is able to demonstratethe quantum information processing in neural architecture,and has the potential to illustrate the dynamics of neuronsin the human brain when viewed from the QM perspective. Itcan be described mathematically that the storage capacity isexponentially increased. This model applies a fuzzy inferencebased Hebb-like learning to a weight learning process forsatisfying feathers of QM. However, QHAM is only able toperform an autoassociation. Thus, in this paper, we extendQHAM to multidirectional association model as QMAM thatholding the features of QM, and present the mathematicalproofs compliant with QMAM.

Online learning is significant in modern systems, such asin processing huge amounts of data. In regards to associativememory models, the time difference Hebb association [26],and neural graph based iterative learning [27] are simple, buteffective approaches. Chartier et al. [28], [29] proposed aself-convergent iterative learning algorithm with a nonlinearoutput function, which does not require batch process, orreference to previous state. This iterative learning algorithm isable to progressively develop a resonance state in the weightmatrices without overlearning, and its online learning processexplicitly incorporates the desired information. We introducethe learning algorithm by Chartier et al. [28], [29] based onits superiority to QMAM to develop a IQMAM.

III. QUANTUM MECHANICS FOR ASSOCIATIVE MEMORY

As mentioned in Section II, Rigatos and Tzafestas [11]introduced the simple and effective QM based approach forassociative memory model as QHAM, and presented math-ematical proofs for satisfying superposition and unitarity.However, QHAM and its mathematical proofs are limited as anautoassociation and batch learning. In this paper, we introducequantum-inspired heteroassociation models with batch and


MASUYAMA et al.: QMAM WITH A SELF-CONVERGENT ITERATIVE LEARNING 3

Fig. 1. Three-layer MAM with weight matrices in αth and βth layers.

incremental learning and its mathematical proofs to satisfythe features of QM. In regards to the heteroassociation model,it can be divided into a bidirectional model and a multidi-rectional model. Typically, a multidirectional model can bedecomposed into multiple bidirectional models that is shownin Fig. 1. Therefore, it can be regarded that the weightconnections in a multidirectional model are fundamentallyequivalent to connections in a bidirectional model. In otherwords, if the features of QM are satisfied in a bidirectionalmodel, it also satisfying in a multidirectional model.

It is regarded that superposition and unitarity are onenoteworthy factors of QM. Superposition can be detailed asexistence of “multiple states” in quantum system. On the otherhand, unitary transformation can be explained the “evolution”in a closed quantum system. In quantum-inspired associativememory, the decomposed weight matrices satisfies superpo-sition based on fuzzy inference, and the rotations betweeneigenvector spaces in decomposed weight matrices satisfiesthe unitary transformation.

A. Hebb-Like Learning Based on Fuzzy Inference

The elements wi j of the weight matrix W are consideredas fuzzy variables, and the increase/decrease operation for theelement wi j is performed based on the following: [30].

1) Increase

IF wi j (k) is A1 THEN wi j (k+1) is A2IF wi j (k) is A2 THEN wi j (k+1) is A3

...IF wi j (k) is An−1 THEN wi j (k+1) is An.

2) Decrease

IF wi j (k) is A2 THEN wi j (k+1) is A1IF wi j (k) is A3 THEN wi j (k+1) is A2

...IF wi j (k) is An THEN wi j (k+1) is An−1

where Ai (i = 1, n) denotes the fuzzy subsets in which theuniverse of discourse of the variable wi j is partitioned. Thefuzzy sets Ai based on a triangular fuzzifier are depicted asFig. 2. The derivation of the fuzzy relational matrices Ri

n andRd

n apply the min t-norm. The max-min inference is utilizedand the defuzzifier being the center of an average one. Theabove-mentioned fuzzy rules can be considered as the Hebb-like learning with the fundamental memory vectors xk and yk

as follows:IF sgn

(xi

k y jk

)> 0 THEN increase wi j

IF sgn(xi

k y jk

)< 0 THEN decrease wi j .

The weight learning process with the above-mentionedfuzzy learning rule is a consequence of a fuzzy weight matrix.Thanks to the fuzzy learning, not a Hebb learning, the weightmatrix W of associative memory can be decomposed andsatisfied unitarity in the decomposed weight matrices W̄ ,which are detailed in Sections III-D and III-E.

B. Basis of Quantum Mechanics

In QM, the vector |ψ(t)〉 represents the state of an isolatedquantum system Q, and it satisfies Schrödinger’s diffusionequation in a Hilbert space [31]

i h̄d

dt|ψ(t)〉 = Hψ(t). (1)

where H provides total energy of a particle as H = (p2/2m)+V(x) as the Hamiltonian operator. Equation (1) representsthe waveform with a probability density which diffusing themomentum p, being proportional to |ψ(x,t)|2, of a particleat position x on time t . V denotes an external potential asV = −(p2/2m)+ E , where E represents the eigenvalue of H .For V = 0 (or constant value), the solution of (1) can beregarded as a superposition of plane waves as follows:

|ψ(x,t)〉 = ei(px−Et)/h̄. (2)

where i represents an imaginary unit, x denotes the particleposition, and h̄ represents Plank’s constant.

The observation probability between particle positions x andx + dx at time t is defined as P(x)dx = |ψ(x,t)|2, which mustbe satisfied the condition as

∫ ∞−∞ |ψ(x,t)|2dx = 1. Here, the

average particle position x is defined as follows:

〈x〉 =∫ ∞

−∞P(x)dx =

∫ ∞

−∞(ψ∗xψ)dx . (3)

where exponential ∗ denotes the conjugate matrix operation.The wave function ψ(x,t), which is a set of orthonormaleigenfunctions, is analyzed in a Hilbert space as follows:

ψ(x,t) =∞∑

m=1

cmψm . (4)

here, cm denotes the probability of particle position x at time t ,which is defined as cm = ∫ ∞

−∞ ψmψ∗dx . The eigenvectors

and eigenvalues of orthonormal eigenfunction at position x isdescribed as xψm = amψm , here ψm is the eigenvector andam is the related eigenvalue. Based on (1) and (2), the averageparticle position is defined as follows:

〈x〉 =∞∑

m=1

‖cm‖2am . (5)

where ‖cm‖2 represents the probability of particle position x ,provided from the associated eigenvalue am . Here, am isselected with probability P ∝ ‖cm‖2.



Fig. 2. Fuzzy subsets in αth layer for weight matrix.

Similarly, observation probability between particle positionsy and y +dy at time t can be described as P(y)dy = ∣

∣ψ(y,t)∣∣2,

which can be obtained, as follows:

〈y〉 =∞∑

n=1

‖cn‖2bn. (6)

where ‖cn‖2 represents the probability of particle position y,which can be described as cn = ∫ ∞

−∞ ψnψ∗dy.

C. Similarity between Quantum Mechanics andFuzzy Inference

Let the variables x and y are belonging to the universeof discourse that can be quantized into an infinite numberof fuzzy subsets Ai , A−i (i = 1, 2, . . . ,∞) and B j , B− j

( j = 1, 2, . . . ,∞), respectively. As an example, Fig. 2 showsthe fuzzy subsets in αth layer. The fuzzy subsets satisfy thefollowing properties.

1) Summation of membership grade is unity as follows.

a) αth → βth layer∞∑

m=1

μAm (x) +∞∑

m=1

μA−m (x) = 1. (7)

b) βth → αth layer∞∑

n=1

νBn(y) +∞∑

n=1

νB−n(y) = 1. (8)

2) The center of fuzzy subsets am and bn are belonging toAm and Bn , respectively.

3) The average values of variables x and y are given bythe following.

a) αth → βth layer

〈x〉 =∞∑

m=1

μAm (x)am(W T

ij ≥ 0). (9a)

〈x〉 =∞∑

m=1

μA−m (x)a−m(W T

ij < 0). (9b)

b) βth → αth layer

〈y〉 =∞∑

n=1

νBn(y)bn (Wij ≥ 0) (10a)

〈y〉 =∞∑

n=1

νB−n(y)bn (Wij < 0). (10b)

Rigatos and Tzafestas presented details of relationshipsbetween fuzzy inference and QM from the above-mentionedproperties as follows [11], [32].

1) The membership grades μAm (x), μA−m (x) correspond to‖cm‖2, ‖c−m‖2.

2) The centers of the fuzzy subsets am in Am and a−m inA−m depict eigenvalues am and a−m of the position x ,respectively.

3) Fuzzy subsets Am and A−m is in tune with probabilityof the particle position, given by |ψx |2.

4) The membership grades μAm (x) and μA−m (x) offuzzy subsets Am and A−m satisfy the condition∑∞

m=1 μAm (x) + ∑∞m=1 μA−m (x) = 1. This condi-

tion is equivalent to total probability of particleposition that equals unity in quantum system, i.e.,∑∞

−∞ |ψ(x)|2dx = 1.5) The outputs of the quantum system are the eigenvalue

Am and A−m that are associated with the eigenvectorsψm and ψ−m , with probabilities P ∝ ‖cm‖2 and P ∝‖c−m‖2, respectively. In the fuzzy system, the centers ofthe fuzzy sets Am and A−m , which becomes the outputof the fuzzy system with probability P ∝ μAm (x) andP ∝ μA−m (x), respectively.

6) The equations of particle wave can be defined as ortho-normalized vectors ψm and ψ−m in a Hilbert space,while the fuzzy variable can be defined as a vector ina eigenvector space which is transitioned on the fuzzysubsets Am and A−m , respectively.

Similarly, centers of fuzzy subsets bn in Bn and b−n in B−n

can be met with the relationships mentioned previously.

D. Superposition Existence in the Weight Matrix

The decomposition process of weight matrices between αthand βth layers will be shown in this section.

1) αth → βth Layer: Based on element w j i from weightmatrix W T , in which the weight belongs to adjoining fuzzysubsets Ai and Ai+1 (shown in Fig. 2) due to the samewidth triangular function. Here, the corresponding centers ofthe fuzzy subset are represented as ai

j i for Ai and ai+1j i for

Ai+1, respectively. Furthermore, the associated membershipgrades are represented as μ j i = μAi and 1 − μ j i = μAi+1 ,respectively. Thus, the element w j i can be defined by theset of {μ j i , a Ai

j i } and {1 − μ j i , a Ai+1j i }, respectively. From the

membership grades μ j i and 1 − μ j i , the matrices for weightdecomposition process can be defined, namely, the weight



TABLE I

POTENTIAL COMBINATIONS OF CENTERS OF FUZZY SUBSETS WITH MEMBERSHIP GRADES FOR W T

TABLE II

POTENTIAL COMBINATIONS OF CENTERS OF FUZZY SUBSETS WITH MEMBERSHIP GRADES FOR W

matrix W T can be decomposed into the set of superpositionmatrices W̄ T

i

(i = 1, 2, . . . , 2N M

)as in Table I. It can be

summarized as follows:

W T =2M N∑

i=1

μi W̄ Ti . (11)

This can be regarded as superposition. Note that the fuzzysubsets A1, A2, . . . , Am−1, Am and A−1, A−2, . . . , A−m+1,A−m in Fig. 2 are within the universe of discourse of theelements w j i of weight matrix W T . In addition, as it is thesame spread and satisfies the strong fuzzy partition equalities,the condition

∑mi=1 μAm (x)+

∑mi=1 μA−m (x) = 1 is maintained.

2) βth → αth Layer: Similarly with αth → βth layer, theelement wi j of weight matrix W is considered. The wi j can

be represented using set of {νi j , bB ji j } and {1 − νi j , ν

B j+1i j },

respectively. Therefore, this can be decomposed the weightmatrix W into the set of superposition matrices W̄ j ( j =1, 2, . . . , 2M N ) as Table II. This results with the weight matrixW , which is described as follows:

W =2N M∑

j=1

ν j W̄ j . (12)

W = ν11 + ν12 + · · · + νmn

N M · 2N M−1

⎡

⎢⎢⎢⎢⎣

bBi11 bBi

12 . . . bBi1n

bBi21 bBi

22 . . . bBi2n

......

. . ....

bBim1 bBi

m2 . . . bBimn

⎤

⎥⎥⎥⎥⎦

+ ν12 + · · · + νmn − ν11 + 1

N M · 2N M−1

⎡

⎢⎢⎢⎢⎣

bBi+111 bBi

12 . . . bBi1n

bBi21 bBi

22 . . . bBi2n

......

. . ....

bBim1 bBi

m2 . . . bBimn

⎤

⎥⎥⎥⎥⎦

+ ν11 + ν13 + · · · + νmn − ν12 + 1

N M · 2N M−1

×

⎡

⎢⎢⎢⎢⎢⎣

bBi11 bBi+1

12 . . . bBi1n

bBi21 bBi

22 . . . bBi2n

......

. . ....

bBim1 bBi

m2 . . . bBimn

⎤

⎥⎥⎥⎥⎥⎦

+· · ·+ −ν11 − ν12 · · · − νmn + N M

N M · 2N M−1

×

⎡

⎢⎢⎢⎢⎢⎣

bBi+111 bBi+1

12 . . . bBi+11n

bBi+121 bBi+1

22 . . . bBi+12n

......

. . ....

bBi+1m1 bBi+1

m2 . . . bBi+1mn

⎤

⎥⎥⎥⎥⎥⎦

(13)

Let us suppose a m by n weight matrix W for anexample. Elements wmn of weight matrix W are consid-ered as fuzzy variables. The decomposed weight matricesW̄k (k = 1, 2, . . . , 2mn) which is shown in (13) can bedescribed by membership grades νmn and its center of fuzzysubsets bBi

mn . The elements in decomposed matrices, indicatedby Ni and associated ‖L1‖ norm, are calculated. Each L1 normis then split by number of elements in matrices Ni . It can beregarded as superposition.

Lemma 1: The ‖L1‖ norm of the matrices Ni are definedas

∑Mi=1

∑Nj=1 |νi j |. The result of the dividing ‖L1‖ norm in

the number of elements in matrices Ni , i.e., N M and 2(N M−1)

(M and N represent the number of neurons in αth and βthlayers, respectively), maintains the unity

1

N M · 2(N M−1)

M∑

i=1

N∑

j=1

|νi j | = 1. (14)



Proof:The two matrices Ni and N j by ν(wi j ) and 1 − ν(wi j ),

respectively, can be obtained due to the fuzzy subsets areconsisted by the triangular form. The summation of thecorresponding L1 norms of Ni and N j with normalizing usingnumber of elements is equivalent unity. Here, the normaliza-tion procedure is utilized obtaining the membership grades forweight matrices W̄i .

E. Unitarity Representation in Fuzzy Inference

The interpretation of“increase” and “decrease” operatorsin fuzzy inference from QM perspective, corresponding toheteroassociation model based on [33] is presented. The fuzzyrelational matrices Ri

m and Rdm (αth layer), and Ri

n and Rdn

(βth layer) are derived based on t-norm, respectively. Here,the exponentials i and d denote “increase” and “decrease,”respectively. The matrices Ri and Rd satisfy the followingproperties.

1) αth → βth Layer

a) W Tij ≥ 0 :

Am = Rim ◦ Am−1

Am−1 = Rdm ◦ Am . (15a)

b) W Tij < 0 :

A−m = Ri−m ◦ A−m+1

A−m+1 = Rd−m ◦ A−m . (15b)

2) βth → αth Layer3) Wij ≥ 0 :

Bn = Rin ◦ Bn−1

Bn−1 = Rdn ◦ Bn. (16a)

4) Wij < 0 :B−n = Ri−n ◦ B−n+1

B−n+1 = Rd−n ◦ B−n. (16b)

Theorem 1: The rule based “increase” and “decrease” fuzzyrelational operators satisfy the unitarity.

Proof:1) αth → βth Layer: The matrices Ri (W T

ij ≥ 0) and R−i

(W Tij < 0), which are utilized as the "increase" and "decrease"

operators, meet the following properties, respectively.

1) Increase Mode:

a) W Tij ≥ 0 :Ah = Ri

h−1 ◦ Ah−1 (h = 2, 3, . . . ,m). (17a)

b) W Tij < 0 :

A−h = Rd−h+1 ◦ A−h+1 (h = 2, 3, . . . ,m). (17b)

2) Decrease mode:

a) W Tij ≥ 0 :Ah−1 = Rd

h−1 ◦ Ah (h = 2, 3, . . . ,m). (18a)

b) W Tij < 0 :

A−h+1 = Rd−h+1 ◦ A−h (h = 2, 3, . . . ,m). (18b)

Both above-mentioned modes, “◦” represents the max-mincomposition. Substituting Ah−1 = Rd

h−1 ◦ Ah for Ah =Ri

h−1 ◦ Ah−1, then Ah = Rih−1 ◦ (Rd

h−1 ◦ Ah) is derived.In the same manner, A−h = Ri−h+1 ◦ (Rd−h+1 ◦ A−h) isobtained. In addition, Ah = (Ri

h−1 ◦ Rdh−1) ◦ Ah and A−h =

(Ri−h+1 ◦ Rd−h+1) ◦ A−h can be derived, respectively, fromthe associativity of max-min operator. Thus, the followingconditions are satisfied:

{(Ri

h−1 ◦ Rdh−1

) = I (19a)(Ri

−h+1 ◦ Rd−h+1

) = I (19b)

Similarly, substituting Ah = Rih−1◦Ah−1 for Ah−1 = Rd

h−1◦Ah , and A−h = Ri−h+1 ◦ A−h+1 for A−h+1 = Rd−h+1 ◦ A−h ,Ah−1 = (Rd

h−1 ◦ Rih−1) ◦ Ah−1 and A−h+1 = (Rd

−h+1 ◦Ri

−h+1)◦ A−h+1 are derived, respectively. Thus, the followingconditions are satisfied:

{(Rd

h−1 ◦ Rih−1

) = I (20a)(Rd−h+1 ◦ Ri−h+1

) = I. (20b)

Furthermore, Mandani’s inference system [34] is appliedto define the matrices Ri

h−1 and Rdh−1, Ri

−h+1 and Rd−h+1,

respectively. Therefore, the followings are obtained:⎧⎨

⎩

Rdh−1 = (

Rih−1

)T (21a)

Rd−h+1 = (Ri−h+1

)T. (21b)

Finally, the following relationships can be derived from(19a) and (20a), (19b) and (20b), respectively, whichshow:

⎧⎨

⎩

(Rd

h−1

)−1 = (Ri

h−1

)T (22a)(Rd

−h+1

)−1 = (Ri

−h+1

)T. (22b)

Therefore, the fuzzy relational matrices Rh−1 and R−h+1,with "increase"/"decrease" operators for αth → βth layer,satisfy the unitarity.

2) βth → αth Layer: Similar to the αth → βthlayer procedure, the fuzzy relational matrices R j

(Wij ≥ 0

)

and R− j(Wij < 0

)satisfy the following properties,

respectively:{(

Rdg−1

)−1 = (Rig−1

)T (23a)(Rd−g+1

)−1 = (Ri−g+1

)T (23b)

where g (g = 2, 3, . . . , n) denotes the number of neurons inβth layer.

Thus, the fuzzy relational matrices Rg−1 and R−g+1 for βth→ αth layer satisfy the unitarity.

F. Transition of Eigenvector Spaces via Unitary Operator

The following theorem shows that the existenceof unitarity in the decomposed weight matrices W̄ T

and W̄ .



Theorem 2: Transition from the eigenvector spaces, definedby the decomposed weight matrices W̄ T and W̄ , via unitaryoperator.

Proof:1) αth → βth Layer: Given wi , xi , yi , zi and w j , x j ,

y j , z j are elements of the vector that span the eigenvec-tor spaces associated by the weight matrices W̄ T

i and W̄ Tj ,

respectively, and assuming the memory vectors p for eachspace are described as p = [pwi , pxi , pyi , pzi ]T and p =[pw j , px j , py j , pz j ]T , respectively. Furthermore, the transi-tion from the eigenvector spaces W̄ T

i with{wi , xi , yi , zi } toW̄ T

j with {w j , x j , y j , z j } is described by the matrix R. Here,the elements of the weight and memory vectors give thefollowing conditions; pW̄ T

i= pwiwi + pxi xi + pyi yi + pzi zi and

pW̄ Tj

= pw jw j + px j x j + py j y j + pz j z j . Then, the followingequations are defined:

pW̄ Ti

= R · pW̄ Tj

⇒

⎡

⎢⎢⎣

pwi

pxi

pyi

pzi

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

wiw j wi x j wi y j wi z j

xiw j xi x j xi y j xi z j

yiw j yi x j yi y j yi z j

ziw j zi x j zi y j zi z j

⎤

⎥⎥⎦

⎡

⎢⎢⎣

pw j

px j

py j

pz j

⎤

⎥⎥⎦. (24)

Similar with above-mentioned, the transition from the eigen-vector spaces from pW̄ T

jto pW̄ T

ican be described as pW̄ T

i=

Q · pW̄ Tj

. Since “dot” products are commutative, one obtains

Q = R−1 = RT . Thus, transition from the eigenvector spacesW̄ T

j to W̄ Ti can be represented by unitary operators as follows:

Q R = RT R = R−1 R = I. (25)

2) βth → αth Layer: Given w j , x j , y j , z j and wi , xi , yi , zi

are the elements of the vector that span the eigenvector spacesassociated by the weight matrices W̄ j and W̄i , respectively, andassuming the memory vectors p for each space are describedas s = [sw j , sx j , sy j , sz j ]T and s = [swi , sxi , syi , szi ]T , respec-tively. Furthermore, the transition from the eigenvector spacesW̄ j with{w j , x j , y j , z j } to W̄i with {wi , xi , yi , zi } is describedby the matrix V . Here, the elements of the weight and memoryvectors give the following conditions; sW̄ j

= sw jw j + sx j x j +sy j y j + sz j z j and sW̄i

= swiwi + sxi xi + syi yi + szi zi . Then,the following equations can be defined:

sW̄i= V · sW̄ j

⇒

⎡

⎢⎢⎣

sw j

sx j

sy j

sz j

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

w jwi w j xi w j yi w j zi

x jwi x j xi x j yi x j zi

y jwi y j xi y j yi y j zi

z jwi z j xi z j yi z j zi

⎤

⎥⎥⎦

⎡

⎢⎢⎣

swi

sxi

syi

szi

⎤

⎥⎥⎦. (26)

Similar to above-mentioned, the transition from the eigen-vector spaces from sW̄i

to sW̄ jcan be described as sW̄ j

=U · sW̄i

. Since “dot” products are commutative, one obtainsV = U−1 = U T . Thus, the transition from the eigenvectorspaces W̄i to W̄ j can be represented by unitary operators asfollows:

V U = U T U = U−1U = I. (27)

IV. FUNDAMENTAL STRUCTURE OF QUANTUM-INSPIRED

MULTIDIRECTIONAL ASSOCIATIVE MEMORY

In this section, the equations of QMAM and IQMAMbetween the αth and the βth layers are presented. Here, theweight matrices in QMAM and IQMAM will be satisfied withsuperposition and unitarity. Throughout Sections V-A and V-B,the following definitions are applied; {x (k)(1), x (k)(2) , . . . , x (k)(L)}, fork = 1, 2, . . . , represents the bipolar stored pattern pairs,the total number of stored patterns is k, and L representsthe number of layers in the model. W , U and V are theweight matrices of QMAM and IQMAM, respectively. Theexponential T represents the transpose operation. α and βdenote the layer number. M and N indicate number ofneurons in the αth and βth layers, respectively. The stateof neuron in each layer will be updated until reaching anarbitrary condition. In addition, x is calculated by Gram-Schmidt orthogonalization from fundamental memory vec-tors, according to a1 = A1/‖A1‖(p = 1), bp = Ap −∑k−1

i=p−1 (ai , Ai ) ai and ap = bp/‖bp‖ (2 ≤ p ≤ k),where a and b denote the orthonormalized vector and theorthogonalized vector, respectively.

A. Fundamental Structures of QMAM

The equations of QMAM are shown.

1) αth → βth layer⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

X (k)j (α) =M∑

i=1

W (αβ)Ti j x (k)i(α) (28a)

x (k)j (β) = sgn

⎛

⎜⎜⎝

L∑

α=1α =β

X (k)j (α)

⎞

⎟⎟⎠ (28b)

2) βth → αth layer⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

X (k)i(β) =N∑

j=1

W (αβ)i j x (k)j (β) (29a)

x (k)i(α) = sgn

⎛

⎜⎜⎝

L∑

β=1β =α

X (k)i(β)

⎞

⎟⎟⎠ (29b)

where, sgn(·) denotes a signum function. The weightmatrices W (αβ)T and W (αβ) are as follows:

W (αβ)T = 1

k

k∑

g=1

x (g)T(β) x (g)(α) (30)

W (αβ) = 1

k

k∑

g=1

x (g)T(α) x (g)(β) (31)

B. Fundamental Structures of IQMAM

The equations of IQMAM based on Chartier algorithm[MAM with a self-convergent iterative learning (IMAM)] [29]consists a self-convergent iterative learning algorithm witha nonlinear output function. According to [29], the learn-ing algorithm is not required any preprocessing to stored



TABLE III

MEMORY CAPACITY CONDITIONS WITH BIPOLAR NEURON REPRESENTATION AND RANDOM PATTERN CONFIGURATION

patterns such as normalization or orthogonalization. Here,orthonormalization will be applied to IQMAM to satisfyingthe conditions of quantum-inspired model. Thus, we assumethat the abilities of IQMAM show the similar results withIMAM though the preprocessing is applied as follows:

1) αth → βth layer⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

X (k)j (α) =M∑

i=1

Uij x (k)i(α) (32a)

x (k)j (β) =L∑

α=1α =β

[(δ + 1)X (k)j (α)(t) − δX (k)3j (α)(t)

](32b)

2) βth → αth layer⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

X (k)i(β) =N∑

j=1

Vij x (k)j (β) (33a)

x (k)i(α) =L∑

β=1β =α

[(δ + 1)X (k)i(β)(t) − δX (k)3i(β)(t)

](33b)

where, δ is a general output parameter, t denotes thenumber of iterations over the network. The weightmatrices U and V are as follows:U(t+1) = U(t) + η(x(β)(0)−x(β)(t))(x(α)(0)+x(α)(t))

T

(34)

V(t+1) = V(t) + η(x(α)(0) − x(α)(t))(x(β)(0) + x(β)(t))T

(35)

where η denotes a learning parameter. x(α)(0) and x(β)(0)represent the initial input information that are selectedrandomly based on a uniform distribution. Those inputsare iterated t times through the network. Since thelearning parameter η satisfies the following condition,the convergent of network is guaranteed [29]:

η <1

2(1 − 2δ)Max[M, N] , δ = 1/2. (36)

where δ is a general output parameter.

V. EXPERIMENTS

For the experiments, we compared with MAM [14], Quick-Learning MAM (QL-MAM) [15], IMAM [29], QMAM, andIQMAM from the viewpoint of memory capacity and noisetolerance, respectively. MAM is a basic model of multidi-rectional associative memory. The QL-MAM is applied theHebb learning and the pseudo relaxation learning to MAM.

Through this section, we utilize three-layer and five-layerMAM model for comparison. The random bipolar patterns areset as stored data to models.

The learning parameters of QL-MAM and IMAM/IQMAMare set as follows; QL-MAM; the relaxation factor λ is 1.9and the normalizing constant ξ is 0.1, IMAM/IQMAM; thegeneral output parameter σ is 0.4, the learning parameterη is 1.5, and the number of iterations is 3000. Further-more, the order of association process is performed from thelayer 1 to 3 (3-layers model) or 5 (5-layers model) untilreaching an arbitrary condition. The initial conditions of layersare described in Sections V-A and V-B.

A. Memory Capacity

Memory Capacity is a significant factor for performancein associative memory. The memory capacity, in general, isresponsive to the number of neurons, e.g., the memory capacityof model is directly affected by the magnitude of numberof neurons. Moreover, the differences in number of neuronsbetween layers is another significant factor for the performanceof memory capacity. Therefore, in regards to memory capacity,the two types of conditions are considered; the constantnumber of neurons being set in layers, and different numberof neurons applied to layers. From the above-mentioned twoconditions, it can be evaluated that the sensitivity of thememory capacity from the viewpoint number of neurons.Here, the layer 1 is assigned with desired information whileothers are assigned the random bipolar patterns as the initialconditions.

At first, we conducted the experiment with the constantnumber of neurons in each layer as Table III. In Fig. 3, dueto the average number of neurons are big, conditions (I-b)and (II-b) indicate the superior recall rate as compared toconditions (I-a) and (II-a), respectively. From the viewpointof number of layers, while the layers increase, the memorycapacity decrease. In addition, the initial condition of layersare set with the random bipolar states which can be regarded asnoise information. Thus, compared with 3-layers and 5-layersmodels, 3-layers model shows the better results than 5-layersmodel. In terms of batch learning models in Fig. 3, QMAMshows the compatible results in condition II, and especiallyshows the superior results with the large number of patternsis stored in condition I. Compared with IMAM and IQMAM,IMAM shows the outstanding results in condition I. However,IMAM is influenced by the number of layers dramatically.In contrast, IQMAM shows the stable results which arefollowing the trend of other models.

Next, we conducted the experiment with a different numberof neurons in layers, which is summarized in Table IV.



Fig. 3. Memory capacity results using constant number of neurons (conditions I and II). (a) Condition (I-a). (b) Condition (I-b). (c) Condition (II-a).(d) Condition (II-b).

TABLE IV

MEMORY CAPACITY CONDITIONS WITH BIPOLAR NEURON REPRESENTATION AND RANDOM PATTERN CONFIGURATION IN EACH LAYER

Fig. 4. Results of Memory Capacity With Different Number of Neurons (conditions III and IV). (a) Condition (II-a). (b) Condition (II-b). (c) Condition (IV-a).(d) Condition (IV-b).

In general, the recall process from the layer with small numberof neurons to the layer with large number of neurons isseverer than the opposite direction. Thus, the recall rate ofconventional models with conditions (III-b) and (IV-b) showthe superior results than the conditions (III-a) and (IV-a),respectively. However, this feature is considered as the insta-bility for differences in number of neurons in layers.

In Fig. 4, though the batch/incremental learning models, theconventional models have the large differences in recall ratebetween conditions (a) and (b) in Table IV. IMAM, especially,shows the quite large differences under the conditions inTable IV. Under the same conditions as above-mentioned, incontrast, QMAM and IQMAM show the similar recall rates.Thus, these models have the stability against the differencesin number of neurons in the layers. In case of the realenvironment, information association will be performed withthe multi-modal information such as verbal, object and so on.Typically, each modal is composed by the different number ofdimensions. Thus, the resistance to the dimensional differenceis significant factor for associative memory.

B. Noise Tolerance

In general, the “noise” is divided to two conditions in theassociation models; one is due to the resemblance between

the stored information, another is due to the stored informationcontain the wrong information. In this experiment, we considerthe latter case. Thus, the initial condition of layers is definedas follows; the number of stored pairs is set as 30. The salt andpepper noise (0–100 [%]) is added to initial state in layer 1,and the random bipolar patterns are set in other layers in thenetwork.

Table V shows the experimental conditions for noise tol-erance. As will be described again, the number of lay-ers is regarded as source of noise. Thus, compared withFigs. 5 and 6, Fig. 5 shows better performance between cor-responding conditions although 3-layers and 5-layers modelshave the same average number of neurons. We first focused inFig. 5. In Fig. 5(a), QMAM and IQMAM show the superiorrecall reliability than other models. Though the number ofneurons in layers is different (i.e., conditions (b) and (c)in Table V), QMAM and IQMAM maintain the high recallreliability, and the difference of recall rate in between condi-tions (b) and (c) is smaller than other models. Similar with3-layers model, Fig. 6 shows QMAM and IQMAM show thesuperior recall reliability and stability to the different numberof neurons in each layer. From Figs. 5 and 6, incidentally,it is regarded that the noise tolerance of IMAM is quitelow.



TABLE V

NOISE TOLERANCE CONDITIONS

Fig. 5. Results of Noise Tolerance in three-layer model (condition V). (a) Condition (V-a). (b) Condition (V-b). (c) Condition (V-c).

Fig. 6. Results of noise tolerance in five-layers model (condition VI). (a) Condition (VI-a). (b) Condition (VI-b). (c) Condition (VI-c).

Based on the results presented, it can be considered thatQMAM and IMAM have the compatible or superior abilitiesthan conventional models. Noteworthy, the proposed modelshave the outstanding stability in association process.

VI. CONCLUSION

In this paper, quantum-inspired multidirectional associationmodel with an one-shot learning and a self-convergent iterativelearning, namely QMAM and IQMAM, have been presented.Specifically, we have introduced quantum-inspired multidirec-tional association model. The simulation experiments show theproposed models have the superior stable abilities to memorycapacity and recall reliability. In future work, we will integratethe associative memory with other functions in human brain,such as emotion. We will also integrate the neural associativememory and emotional model to define a type of emotionalassociative memory from the QM perspective.

ACKNOWLEDGMENT

The authors would like to thank the University of Malaya(Fellowship Scheme) for the scholarship.

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Naoki Masuyama (S’12–M’16) received the B.Eng.degree from Nihon University, Tokyo, Japan, in2010, the M.Eng. degree from Tokyo MetropolitanUniversity, Tokyo, in 2012, and the Ph.D. degreefrom the Faculty of Computer Science and Infor-mation Technology, University of Malaya, KualaLumpur, Malaysia, since 2016.

He is currently a Post-Doctoral Research Fellowwith the Faculty of Computer Science and Informa-tion Technology, University of Malaya. His currentresearch interests include associative memory, clus-

tering/classification, and human–robot interaction.

Chu Kiong Loo (SM’14) received the B.Eng. degree(Hons.) in mechanical engineering from the Uni-versity of Malaya, Kuala Lumpur, Malaysia, andthe Ph.D. degree from Universiti Sains Malaysia,George Town, Malaysia.

He was a Design Engineer in various indus-trial firms and is the founder of the AdvancedRobotics Laboratory, University of Malaya. He hasbeen involved in the application of research intoPeruss Quantum Associative Model and PribramsHolonomic Brain Model in humanoid vision

projects. He is currently a Professor of Computer Science and InformationTechnology with the University of Malaya. He has led many projects fundedby the Ministry of Science in Malaysia and the High Impact ResearchGrant from the Ministry of Higher Education, Malaysia. His current researchinterests include brain-inspired quantum neural networks, constructivism-inspired neural networks, synergetic neural networks, and humanoid research.

Manjeevan Seera (M’11-SM’15) received theB.Eng. degree (Hons.) in electronics and electri-cal engineering from the University of Sunderland,Sunderland, U.K., in 2007, and the Ph.D. degreein computational intelligence from Universiti SainsMalaysia, George Town, Malaysia, in 2012.

He is currently an Adjunct Research Fellow withthe Swinburne University of Technology SarawakCampus, Kuching, Malaysia. His current researchinterests include soft computing, pattern recogni-tion, fault detection and diagnosis, and human–robot

interaction.

Naoyuki Kubota (S’95–A’97–M’01) received theB.Sc. degree from Osaka Kyoiku University, Kashi-wara, Japan, in 1992, the M.Eng. degree fromHokkaido University, Hokkaido, Japan, in 1994, andthe D.E. degree from Nagoya University, Nagoya,Japan, in 1997.

He joined the Osaka Institute of Technology,Osaka, Japan, in 1997. He joined the Department ofHuman and Artificial Intelligence Systems, Univer-sity of Fukui, Fukui, Japan, as an Associate Profes-sor in 2000. He joined the Department of Mechanical

Engineering, Tokyo Metropolitan University, Hachioji, Japan, in 2004. He wasan Associate Professor from 2005 to 2012, and has been a Professor since2012 with the Department of System Design, Tokyo Metropolitan University,Tokyo, Japan.

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