quantum-inspired bidirectional associative memory … · quantum-inspired bidirectional associative...

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Quantum-Inspired Bidirectional Associative Memory for HumanRobot Communication Naoki Masuyama * and Chu Kiong Loo Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia * [email protected] [email protected] Naoyuki Kubota Department of Systems Design, Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo, 191-0065, Japan [email protected] Received 11 August 2013 Accepted 23 January 2014 Published 30 May 2014 The emerging research area of a quantum-inspired computing has been applied to various ¯eld such as computational intelligence, and showed its superior abilities. However, most existing researches are focused on theoretical simulations, and have not been implemented in systems under practical environment. For humanrobot communication, associative memory becomes essential for multi-modal communication. However, it always su®ers from low memory capacity and recall reliability. In this paper, we propose a quantum-inspired bidirectional associative memory with fuzzy inference. We show that fuzzy inference satis¯es basic postulates of quantum mechanics, but also learning algorithm for weight matrix in associative memory. In addition, we construct a communication system with robot partner using proposed model. This is the ¯rst successful attempt to overcome conventional problems in associative memory model with a robot application. Keywords: Associative memory; quantum computing; fuzzy inference; humanrobot interaction. 1. Introduction Recently, the aging society is one of the inevitable problems in some countries. Especially, the number of elderly people living alone is increasing and they are lacking a chance of communication with other people compared with elderly people who live with family. 1 Thus, these people would increase the probability of the cognitive decline and a high risk of dementia. In order to improve the situation of less communication, some elderly care robot has been developed. 27 International Journal of Humanoid Robotics Vol. 11, No. 2 (2014) 1450006 (22 pages) ° c World Scienti¯c Publishing Company DOI: 10.1142/S0219843614500066 1450006-1 Int. J. Human. Robot. Downloaded from www.worldscientific.com by Dr. Chu Kiong Loo on 06/10/14. For personal use only.

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Page 1: Quantum-Inspired Bidirectional Associative Memory … · Quantum-Inspired Bidirectional Associative Memory for Human–Robot Communication Naoki Masuyama* and Chu Kiong Loo† Faculty

Quantum-Inspired Bidirectional Associative Memory

for Human–Robot Communication

Naoki Masuyama* and Chu Kiong Loo†

Faculty of Computer Science and Information Technology,

University of Malaya, 50603 Kuala Lumpur, Malaysia*[email protected]

[email protected]

Naoyuki Kubota

Department of Systems Design,Tokyo Metropolitan University,

6-6 Asahigaoka, Hino, Tokyo, 191-0065, Japan

[email protected]

Received 11 August 2013

Accepted 23 January 2014Published 30 May 2014

The emerging research area of a quantum-inspired computing has been applied to various ¯eld

such as computational intelligence, and showed its superior abilities. However, most existing

researches are focused on theoretical simulations, and have not been implemented in systemsunder practical environment. For human–robot communication, associative memory becomes

essential for multi-modal communication. However, it always su®ers from low memory capacity

and recall reliability. In this paper, we propose a quantum-inspired bidirectional associative

memory with fuzzy inference. We show that fuzzy inference satis¯es basic postulates of quantummechanics, but also learning algorithm for weight matrix in associative memory. In addition, we

construct a communication system with robot partner using proposed model. This is the ¯rst

successful attempt to overcome conventional problems in associative memory model with arobot application.

Keywords: Associative memory; quantum computing; fuzzy inference; human–robot interaction.

1. Introduction

Recently, the aging society is one of the inevitable problems in some countries.

Especially, the number of elderly people living alone is increasing and they are

lacking a chance of communication with other people compared with elderly people

who live with family.1 Thus, these people would increase the probability of the

cognitive decline and a high risk of dementia. In order to improve the situation of less

communication, some elderly care robot has been developed.2–7

International Journal of Humanoid Robotics

Vol. 11, No. 2 (2014) 1450006 (22 pages)

°c World Scienti¯c Publishing Company

DOI: 10.1142/S0219843614500066

1450006-1

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The issue of social communication has been discussed in sociology, develop-

mental psychology, relevance theory, and embodied cognitive science.8–14 Cogni-

tive psychology has tried to construct a computer that can think.8 In the \society

of mind" theory proposed by Minsky, intelligence is explained as a combination of

multiple simpler elements. He argues that although each agent is itself intelligent,

it is not enough to simply explain what each separate agent does. Rather, it is a

group of agents that can accomplish things.11 Similarly, the theory o®ers insight

into the debate about human communication.12 According to this theory, human

thinking is not simply transmitted, but is an event that is shared between two

people. Such a shared environment is called a mutual cognitive environment.15

Through the communication with humans, the robot can understand personal

preferences, interests and intentions. And by performing mutual cognitive envi-

ronment sharing with human, the robot can provide conversation topics by itself.

In this situation, it is preferable that the selected topics are related to the shared

cognitive environment. Conventionally, topics provided by the robot are selected

based on previous conversations with the human or human behaviors. In our

previous work, we proposed a multi-modal communication system and its

computational intelligence.16 The basic idea of the multi-modal communication

and cognitive environment are similar to this paper. But previous paper is not

considered relevance or continuity of topics. On the other hand, the system in this

paper considered it. Thus, this paper can handle relationships between di®erent

information. If robot provides topics without any relevance or continuity, sharing

a cognitive environment is di±cult. However, if the principles of associative

memory in a shared cognitive environment are applied, the robot can provide more

relevent topics for conversation. As a result, communication between the robot

and human will be more smooth and active.

Various types of associative memory have been proposed. In the early 1980's,

Hop¯eld proposed an auto-associative memory model to store and recall information.

This model, however, su®ers from a lack of memory capacity and noise tolerance. In

the late 1980s, Kosko extended the Hop¯eld model and introduced bidirectional

associative memory (BAM).17 However, the original Kosko BAM also su®ers from

low storage capacity and poor recall reliability. To improve the performance, various

types of implementations have been introduced e.g., added dummy neurons to each

layer,18 and added hidden layer.19 A General model for BAM (GBAM) is de¯ned as a

weight matrix in each layer, and developed an algorithm for learning the asymptotic

stability conditions.20 Thanks to this structure, memory capacity and noise tolerance

are quite improved from Kosko BAM. As another approach, there is a model that

apply quantum mechanics for Hop¯eld model.21–23 This model demonstrates that

quantum information processing in neural structures results in an exponential in-

crease in storage capacity and can explain the extensive memorization and inferen-

cing capabilities of humans. However, it is limited to auto-associative memory.

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The existing models su®er from limited memory capacity and noise tolerance

problems. In this paper, we develop a theory for quantum-inspired bidirectional

associative memory (QBAM). The QBAM result from neural associative memories

if the elements w of the weight matrix W are taken to be fuzzy variables. In this

paper, we show that fuzzy inference satis¯es not only basic postulates of quantum

mechanics, but also learning algorithm for weight matrix in associative memory.

Instead of Hebbian learning, it is assumed that the incremental changes of the

weights are performed with the use of fuzzy inference. It is the ¯rst attempt to

provide an e®ective solution to conventional problems in associative memory.

Moreover, it is one of the ¯rst quantum associative memory models that can be

applied to human–robot Interaction in a real-life environment. In Sec. 2, ¯rst of

all, we show the structure of QBAM. Next, we show similarity between fuzzy

inference and quantum mechanics in QBAM, and proof fuzzy inference satis¯es

two basic postulates of quantum mechanics. In Sec. 3, we present simulation

experiment about memory capacity and noise tolerance of QBAM. In Sec. 4, we

explain computational intelligence technologies and the overall architecture of the

robot partner system using the proposed method. Section 5 presents the experi-

mental results of the potential communication between the robot partner and the

human through object, gesture and voice recognition.

2. Quantum Mechanics for Associative Memory

Superposition and unitarity are the key features of quantum mechanics. Superpo-

sition can be explained as \multiple states", which exist simultaneously in the

quantum system. The evolution of a closed quantum system is described by a unitary

transformation. If we apply fuzzy inference with same width triangular membership

functions in weight matrix, existence of superposition and unitary operation are

satis¯ed. In QBAM, unitarity is satis¯ed by rotations between spaces that are

spanned by the eigenvectors of weight matrices.24 Mathematical proofs are presented

from Secs. 2.5–2.8.

2.1. Structure of quantum-inspired BAM

. X-layer to Y-layer

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. Y-layer to X-layer

Let fX ðkÞ;X ðkÞ; . . . ;X ðkÞg and fY ðkÞ;Y ðkÞ; . . . ;Y ðkÞg, for k ¼ 1; 2; . . . ; be the

bipolar pattern to be stored. k denotes the number of pattern pairs, M and N denote

the number of neurons in X-layer and Y-layer, respectively, W Tij and Wij denote the

weight matrix in X-layer and Y-layer, respectively.

The weight matrix W Tij and Wij are as follows:

. X-layer to Y-layer

W Tij ¼ 1

K

XNj¼1

XMi¼1

vTj ui; ð3Þ

. Y-layer to X-layer

Wij ¼1

K

XMi¼1

XNj¼1

uTi vj ; ð4Þ

where ui and vj are calculated by Gram-Schmidt orthogonalization that according to

a1 ¼ A1=jjA1jjðp ¼ 1Þ, bp ¼ Ap �Pk�1

i¼p�1ðai;AiÞai and ap ¼ bp=jjbpjjð2 � p � kÞ,where A denotes the vector of performing orthogonalization, a and b denote

orthonormalized vector and orthogonalized vector, respectively.

The de¯nition of the weight matrix update is as Eqs. (5) and (6).

. X-layer to Y-layer

WTðtþ1Þij ¼ W

TðtÞij þ

�F ; If y kðstoredÞ

XMi¼1

WTðtÞij x k � 0 and W

TðtÞij � 0

F ; If y kðstoredÞ

XMi¼1

WTðtÞij x k � 0 and W

TðtÞij < 0:

8>>>><>>>>:

WTðtÞij ; Otherwise

8>>>>>><>>>>>>:

ð5Þ

. Y-layer to X-layer

Wðtþ1Þij ¼ W

ðtÞij þ

�F ; If x kðstoredÞ

XNj¼1

WðtÞij y k � 0 and W

ðtÞij � 0

F ; If x kðstoredÞ

XNj¼1

WðtÞij y k � 0 and W

ðtÞij < 0:

8>>>><>>>>:

WðtÞij ; Otherwise

8>>>>>>><>>>>>>>:

ð6Þ

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In Eqs. (5) and (6), t denotes time steps, exponential T denotes transpose. xðkÞðstoredÞ

and yðkÞðstoredÞ denote kth stored pattern, and x ðkÞ and y ðkÞ denote inner state of

X-layer and Y-layer, respectively. M and N denote the number of neurons in

X-layer and Y-layer, respectively. Fð0 < F � 1Þ denotes variation amount in po-

sition for the center of Fuzzy sets. The learning algorithm is referred from a

GBAM.20 This model tried to ¯nd an asymptotic stability condition of learning

algorithm for improving its abilities. The fundamental update algorithm is based

on Hebbian learning. Thus, we considered that it can be applied to the proposed

model.

2.2. Eigenstructure analysis of quantum-inspired BAM

Lemma 1. If the fundamental memory vectors of the associate memory are

chosen to be orthogonal, then they are collinear to the eigenvectors of matrices W T

and W .

Proof. The eigenvectors x and y of matrices W and W T satisfy the following:

The fundamental memory vectors �x and �y are taken to be orthogonal to each

other, i.e., �x Tj �x i ¼ �ði � jÞ and �yT

i �yj ¼ �ðj � iÞ, where � is Kronecker's function. The

weight matrix W and W T are given by W ¼ PMi¼1

PNj¼1 �y

Tj �x i and

W T ¼ PNj¼1

PMi¼1 �x

Ti �yj . Thus, the following holds:

. X-layer to Y-layer

W T �xk ¼1

K

XMi¼1

XNj¼1

�x Ti �yj

( )�xk ¼

1

K

XMi¼1

XNj¼1

�yið�x Ti �xkÞ ¼

1

K�yk

) W T �xk ¼1

K�yk : ð8Þ

. Y-layer to X-layer

W �yk ¼1

K

XNj¼1

XMi¼1

�yTj �x i

( )�yk ¼

1

K

XNj¼1

XMi¼1

�x jð�yTj �ykÞ ¼1

K�xk

) W �yk ¼1

K�xk : ð9Þ

Lemma 2. If the memory vectors of an associate memory are chosen randomly and

the number of neurons N and M are large, then there is high probability for them to be

orthogonal.

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Proof. The normalized internal product of memory vectors xi and xk , yj and yk are

considered

where, h denotes the hth fundamental memory, M denotes the number of neurons in

X-layer, N denotes the number of neurons in Y-layer. For largeM and N , and x ih, x

ik ,

y jh, y

jk randomly chosen from f�1; 1g. It holds EðPjiÞ ¼ 0, EðQijÞ ¼ 0 and

Therefore, EðPjiÞ ¼ 0, EðPji � �PjiÞ2 ¼ ð1=NMÞ2, and EðQijÞ ¼ 0, EðQij � �QijÞ2 ¼ð1=MN Þ2. The Central Limit Theorem (CLT) is applied here. This state:

(i) Consider fPkg a sequence of mutually independent random variables fPkg,which follow a common distribution. It is assumed that Pk has mean �P and

variance �2P , and let P ¼ P11 þ P12 þ � � � þ PNM ¼ PN

j¼1

PMi¼1 Pji . Then, as NM

approaches in¯nity, the probability distribution of the sum random variable P

approaches Gaussian distribution

ðP � NM�PÞffiffiffiffiffiffiffiffiffiNM

p�P

\NMð0; 1Þ: ð12Þ

(ii) Consider fQkg a sequence of mutually independent random variables fQkg, whichfollow a common distribution. It is assumed that Qk has mean �Q and variance

�2Q, and let Q ¼ Q11 þQ12 þ � � � þQMN ¼ PM

i¼1

PNj¼1 Qij . Then as MN approa-

ches in¯nity the probability distribution of the sum random variable Q approa-

ches Gaussian distribution

ðQ �MN�QÞffiffiffiffiffiffiffiffiffiMN

p�Q

\MN ð0; 1Þ: ð13Þ

According to CLT, the probability distribution of the sum variablePN

j¼1PMi¼1 y

jhx

ik ¼ NM

PNj¼1

PMi¼1 Pji and

PMi¼1

PNj¼1 x

ihy

jk ¼ MN

PMi¼1

PNj¼1 Qij . Each of

them follows a Gaussian distribution of center:

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Therefore, for large number of neurons M and N , i.e., for high dimensional spaces

1=NM ! 0, 1=MN ! 0 and the vectors xi and xk , yj and yk will be practically

orthogonal.

Thus, taking into account the orthogonality of the fundamental memories xk ,

yk and Lemma 1, it can be deduced that memory patterns in high dimensional

spaces practically coincide with the eigenvectors of the weight matrices W T and W .

2.3. Hebbian learning through fuzzy inference

. X-layer to Y-layer

In Fig. 1(a), A1;A2; . . . ;Am�1;Am and A�1;A�2; . . . ;A�mþ1;A�m are the fuzzy

subsets in which the universe of discourse of the variable wTij . The sets Ai and A�i are

selected to have the same spread and to satisfy the strong fuzzy partition equalityPmi¼1 �AmðxÞ ¼ 1 and

Pmi¼1 �A�mðxÞ ¼ 1, respectively.

. Y-layer to X-layer

In Fig. 1(b), B1;B2; . . . ;Bn�1;Bn and B�1;B�2; . . . ;B�nþ1;B�n are the fuzzy subsets

in which the universe of discourse of the variable wij . The sets Bj and B�j are selected

to have the same spread and to satisfy the strong fuzzy partition equalityPn

j¼1

�BnðyÞ ¼ 1 andPn

j¼1 �B�nðyÞ ¼ 1, respectively.

The fuzzi¯er is selected to be a triangular one. The main t-norm is used for

derivation of the fuzzy relational matrices Rim and Rd

m (X-layer), Rin and Rd

n (Y-

layer), where exponential i and d denotes \increase" and \decrease", respectively.

(a)

(b)

Fig. 1. The fuzzy set in (a) X-layer and (b) Y-layer.

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These matrices have the following properties:

. X-layer to Y-layer

ðaÞ W Tij � 0 :

Am ¼ Rim � Am�1; Am�1 ¼ Rd

m � Am; ð15aÞðbÞ W T

ij < 0 :

A�m ¼ Ri�m � A�mþ1; A�mþ1 ¼ Rd

�m � A�m: ð15bÞ. Y-layer to X-layer

ðaÞ Wij � 0 :

Bn ¼ Rin � Bn�1; Bn�1 ¼ Rd

n � Bn; ð16aÞðbÞ Wij < 0 :

B�n ¼ Ri�n � B�nþ1; B�nþ1 ¼ Rd

�n � B�n: ð16bÞ

The max-min inference is used, while the defuzzi¯er is a center of average one. The

learning algorithm is inspired from Hebbian learning and in the case of binary

memory vectors xk and yk can be stated as follows:

. X-layer to Y-layer

IF ykXmi¼1

W Tij x

ki � 0 and W T

ij < 0; THEN increase W Tij ; ð17aÞ

IF ykXmi¼1

W Tij x

ki � 0 and W T

ij � 0; THEN decrease W Tij : ð17bÞ

. Y-layer to X-layer

IF xkXnj¼1

Wijykj � 0 and Wij < 0; THEN increase Wij ; ð18aÞ

IF xkXnj¼1

Wijykj � 0 and Wij � 0; THEN decrease Wij : ð18bÞ

The weight matrix update with above fuzzy learning algorithm results by fuzzy

weight matrices. The latter can be decomposed into a superposition of associative

memories.

The whole weight matrices of associative memory W T and W equals a weighted

averaging of the individual weight matrices �W T and �W , i.e.,

. X-layer to Y-layer

W T ¼X2M

i¼1

�i �WTi ; ð19Þ

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. Y-layer to X-layer

W ¼X2N

j¼1

�j �Wj ; ð20Þ

where the non-negative weights �i and �j indicate the contribution of each local

associative memory �WTi and �W j . M and N denote the number of neurons in X-layer

and Y-layer, respectively. We will show a brief explanation of fuzzy value and each

variables in weight matrix in Sec. 2.7. In addition, Rigatos presents more details with

numeric example.22

2.4. Fundamental of quantum mechanics

In quantum mechanics, the state of an isolated quantum system Q is represented

by a vector j ðtÞi in a Hilbert space. This vector satis¯es Schr€odinger's di®usion

equation.24

i}d

dtj ðtÞi ¼ H ðtÞ; ð21Þ

where H denotes Hamiltonian operator that gives the total energy of a particle

(potential plus kinetic energy) H ¼ ðp2=2mÞ þ VðxÞ. The probability to ¯nd the

particle between x and x þ dx at the time instant t, the wave function ðx;tÞ can

be analyzed in a set of orthonormal eigenfunctions in a Hilbert space: ðx;tÞ ¼P1m¼1 cm m. Here, the coe±cients cm is an indication of the probability to describe

the particle's position x at time t by the eigenfunction m.

From Eq. (21), the average position of the particle is found to be

hxi ¼X1m¼1

jjcmjj2am; ð22Þ

where jjcmjj2 denotes the probability that the particle's position be described by

the eigenfunction m. The particle position x is the associated eigenvalue am. The

eigenvalue am is chosen with probability P / jjcmjj2.In the same way, the probability to ¯nd the particle between y and y þ dy at the

time instant t is given by PðyÞdy ¼ j ðy;tÞj2, then derived the following:

hyi ¼X1n¼1

jjcnjj2bn; ð23Þ

where jjcnjj2 denotes the probability that the particle's position be described by the

eigenfunction n.

2.5. Similarity of quantum mechanics and fuzzy inference

Assume that the fuzzy variables x and y belong to a universe of discourse, that

is quantized to an in¯nite number of fuzzy sets Ai;A�iði ¼ 1; 2; . . . ;1Þ and

Bj ;B�jðj ¼ 1; 2; . . . ;1Þ, e.g., the axis of real number R is partitioned to an in¯nite

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number of fuzzy sets with the same space and width. The fuzzy sets have the

following properties:

(a) They satisfy

. X-layer to Y-layer

X1m¼1

�AmðxÞ ¼ 1 ðW Tij � 0Þ; ð24aÞ

X1m¼1

�A�mðxÞ ¼ 1 ðW Tij < 0Þ; ð24bÞ

. Y-layer to X-layer

X1n¼1

�BnðyÞ ¼ 1 ðWij � 0Þ; ð25aÞ

X1n¼1

�B�nðyÞ ¼ 1 ðWij < 0Þ: ð25bÞ

(b) Each fuzzy set Am and Bn are described by its center am, bn and its width F .

(c) The average value of variable x and y will be given by

. X-layer to Y-layer

hxi ¼X1m¼1

�AmðxÞam ðW Tij � 0Þ; ð26aÞ

hxi ¼X1m¼1

�A�mðxÞa�m ðW Tij < 0Þ; ð26bÞ

. Y-layer to X-layer

hyi ¼X1n¼1

�BnðyÞbn ðWij � 0Þ; ð27aÞ

hyi ¼X1n¼1

�B�nðyÞbn ðWij < 0Þ: ð27bÞ

2.6. Fuzzy inference is performed through unitary operators

Theorem 1. The increase and decrease fuzzy operators that were described in the

rule-base are unitary.

Proof.

. X-layer to Y-layer

The fuzzy relational matrices RiðW Tij � 0Þ, R�iðW T

ij < 0Þ used by the increase

and decrease fuzzy operators satisfy the following fuzzy relational equations,

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respectively:

(1) Increase mode

ðaÞ W Tij � 0 :

A2 ¼ Ri1 � A1; A3 ¼ Ri

2 � A2; . . . ;Am ¼ Rim�1 � Am�1; ð28aÞ

ðbÞ W Tij < 0 :

A�2 ¼ Ri�1 � A�1; A�3 ¼ Ri

�2 � A�2; . . . ;A�m ¼ Ri�mþ1 � A�mþ1: ð28bÞ

(2) Decrease mode

ðaÞ W Tij � 0 :

A1 ¼ Rd1 � A2; A2 ¼ Rd

2 � A3; . . . ;Am�1 ¼ Rdm�1 � Am; ð29aÞ

ðbÞ W Tij < 0 :

A�1 ¼ Rd�1 � A�2; A�2 ¼ Rd

�2 � A�3; . . . ;A�mþ1 ¼ Rd�mþ1 � A�m: ð29bÞ

In both cases, \�" denotes the max-min composition. Substituting Am�1 ¼Rd

m�1 � Am in Am ¼ Rim�1 � Am�1 one gets Am ¼ Ri

m�1 � ðRdm�1 � AmÞ, in the same

way, A�m ¼ Ri�mþ1 � ðRd

�mþ1 � A�mÞ, and using the associativity of the max-min

composition yields Am ¼ ðRim�1 � Rd

m�1Þ � Am, A�m ¼ ðRi�mþ1 � Rd

�mþ1Þ � A�m,

respectively, i.e.,

Setting Am ¼ Rim�1 � Am�1 in Am�1 ¼ Rd

m�1 � Am, A�m ¼ Ri�mþ1 � A�mþ1 in

A�mþ1 ¼ Rd�mþ1 � A�m, and using the associativity of the max-min composition

yields Am�1 ¼ ðRdm�1 � Ri

m�1Þ � Am�1, A�mþ1 ¼ ðRd�mþ1 � Ri

�mþ1Þ � A�mþ1, i.e.,

Furthermore, due to the generation of the matrices Rim�1 and Rd

m�1, Ri�mþ1 and

Rd�mþ1 using Mandanis inference system,25 it holds the following relation:

From (31a) and (32a), (31b) and (32b) are deduced following:

Here, (33a) and (33b) are satis¯ed unitary.

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. Y-layer to X-layer

In the same way as X-layer to Y-layer, the fuzzy relational matrices RjðWij � 0Þ,R�jðWij < 0Þ used by the increase and decrease fuzzy operators satisfy fuzzy rela-

tional equations. Therefore, same as (30a) and (30b) to (32a) and (32b), it will be

derived following relationships that are similar with (33a) and (33b):

Therefore, the increase and decrease operators are unitary.

2.7. Existence of superposition in the weight matrix

. X-layer to Y-layer

Assume that weight element wkm of matrix W T , i.e., the element of the kth row

and themth column ofW T . Due to strong fuzzy partition, this weight belongs to two

adjacent fuzzy sets Ai and Aiþ1. The corresponding centers of the fuzzy sets are a ikm

and a iþ1km , and because of the strong fuzzy partition, the associatedmemberships will be

�km ¼ �Aiand 1� �km ¼ �Aiþ1

. Therefore, wkm is described by the sets f�km; aAi

kmg andf1� �km; a

Aiþ1

km g. Taking the possible combinations of the memberships for each

weight, the matrices that have as elements the memberships �km, 1� �km are gener-

ated. Taking the possible combinations of the projections of each weight to the centers

of the adjacent fuzzy sets, the matrices that have as elements, the centers aAi

km and

aAiþ1

km are generated. Using the above, the decomposition of the weight matrixW T into

the set of superimposing matrices �WTi ði ¼ 1; 2; . . . ; 2M Þ.

. Y-layer to X-layer

Assume that weight element wkn of matrixW , i.e., the element of the kth row and the

nth column of W . Due to strong fuzzy partition, this weight belongs to two adjacent

fuzzy sets Bj and Bjþ1. The corresponding centers of the fuzzy sets are bjkn and bjþ1kn ,

and because of the strong fuzzy partition, the associated memberships will be �kn ¼�Bj

and 1� �kn ¼ �Bjþ1. Therefore, wkn is described by the sets f�kn ; bBj

kng and

f1� �kn; �Bjþ1

kn g. Here, same as X-layer to Y-layer, the decomposition of the weight

matrix W into the set of superimposing matrices �Wj ðj ¼ 1; 2; . . . ; 2N Þ.

2.8. Evolution of eigenvector spaces via unitary rotations

It will be shown that the transition between the vector spaces that are associated

with matrices �WTi s and �W js are described by unitary rotations, respectively. These

are stated in the following theorem:

Theorem 2. The rotations between the spaces that are spanned by the eigenvectors

of the weight matrices �WTi and �W j are unitary operators.

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Proof.

. X-layer to Y-layer

Let xi, yi, zi and xj , yj , zj be the unit vectors of the bases which span the spaces

associated with the matrices �WTi and �W

Tj , respectively. Then a memory vector p

can be described in both spaces as p ¼ ðpxi ; pyi ; pzi ÞT and p ¼ ðpxj ; pyj ; pzj ÞT . Tran-sition from the reference system �W

Ti ! fxi; yi; zig to the reference system �W

Tj !

fxj ; yj ; zjg is expressed by the rotation matrix R, i.e., p �WTi¼ R � p�w T

j. Taking the

components of vectors p �WTi

and p �WTj, one gets p �W

Ti¼ pxi xi þ pyi yi þ pzi zi and

p �WTj¼ pxj xj þ pyj yj þ pzj zj . Furthermore it is true that

p �WTi¼ R � p �W

Tj)

pxipyipzi

0@

1A ¼

xixj xiyj xizjyixj yiyj yizjzixj ziyj zizj

0@

1A pxj

pyjpzj

0@

1A: ð35Þ

Similarly, one can obtain the transformation from p �WTi

to p �WTj, i.e., p �W

Ti¼

Q � p �WTj. Since dot products are commutative, one obtains Q ¼ R�1 ¼ RT . There-

fore, the transition from the reference system �WTi to the reference system �W

Tj is

described by unitary operators

QR ¼ RTR ¼ R�1R ¼ I : ð36Þ. Y-layer to X-layer

Let xj , yj , zj and xi, yi, zi be the unit vectors of the bases which span the spaces

associated with the matrices �W j and �W i, respectively. Then a memory vector q can

be described in both spaces as q ¼ ðqxj ; qyj ; qzj ÞT and q ¼ ðqxi ; qyi ; qzi ÞT . Transitionfrom the reference system �W j ! fxj ; yj ; zjg to the reference system �W i ! fxi; yi; zigis expressed by the rotation matrix Q, i.e., q �W j

¼ Q � q�wi. Taking the components of

vectors q �W jand q �W i

, one gets q �W j¼ qxj xj þ qyj yj þ qzj zj and q �W i

¼ qxi xi þ qyi yi þ qzi zi.

Furthermore, it is true that

q �W j¼ Q � q �W i

)qxjqyjqzj

0@

1A ¼

xjxi xjyi xjziyjxi yjyi yjzizjxi zjyi zjzi

0@

1A qxi

qyiqzi

0@

1A: ð37Þ

Similarly, one can obtain the transformation from q �W ito q �W j

, i.e., q �W i¼ R � q �W j

.

Since dot products are commutative, one obtains R ¼ Q�1 ¼ QT . Therefore, the

transition from the reference system �W i to the reference system �W j that is described

by unitary operators

RQ ¼ QTQ ¼ Q�1Q ¼ I : ð38Þ

3. Simulation Experiment

In this section, we compare Kosko BAM (BAM),16,17 GBAM20 and QBAM in terms

of memory capacity and noise tolerance. GBAM is de¯ned as a weight matrix in each

layer, and developed an algorithm for learning the asymptotic stability conditions.

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Fig. 2. One part of pattern pair sets (alphabet, number, image).

(a)

(b)

Fig. 3. The result of memory capacity (the number of maximum stored patterns: K ¼ 256). (a) scope:

K ¼ 1 to 30. (b) K ¼ 1 to 256.

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We prepare the number of stored pattern pair sets K ¼ 256 (alphabet, number,

image and random pattern). We considered that if the correct recall rate for k

pattern pairs was over 90%, k pattern pairs could be stored in memory. Figure 2

shows one part of pattern pair sets. These patterns represent 25 (5 by 5) bipolar

patterns as neuron for X-layer and Y-layer, respectively.

3.1. Memory capacity

Memory capacity is an important element of associative memory performance.

Figure 3(a) is the scope of 1 to 30 in Fig. 3(b). In Fig. 3(a) with GBAM, the recall

rate will be lower than 90% around K ¼ 20, and BAM can be stored quite less

pattern pair sets. On the other hand, in Fig. 3b, the result of QBAM shows over 90%

recall rate with any point of number of pairs K. In addition, the number of stored

patterns will be increased, and the recall rate will be decreased in BAM and GBAM.

(a)

(b)

Fig. 4. The result of noise tolerance. (a) K ¼ 20. (b) K ¼ 75.

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In contrast, the recall rate of QBAM keeps over 90% with any points, and even if the

number of stored patterns is increased, the recall rate can be held. Therefore, we can

regard that QBAM has outstanding memory capacity than BAM and GBAM.

3.2. Noise tolerance

Noise tolerance is another signi¯cant function in associative memory. We measure

the noise tolerance by adding the noises on input data randomly in X-layer. We set

two types of conditions (K ¼ 20, 75). According to Fig. 3(a), BAM cannot memorize

correctly in K ¼ 20, GBAM can be stored in K ¼ 20. QBAM can be stored in both

conditions. In Fig. 4(a), the recall rate of BAM and GBAM are decreased at the point

of 50% noise rate. Considering recall rate, GBAM can be stored correctly with 50%

noise. In contrast, QBAM shows quite high recall rate even if input data contains

high noise rate. From the result in Fig. 3(b), only QBAM can be stored as pattern

pair sets with K ¼ 75. Same as Fig. 4(a), QBAM in Fig. 4(b) shows high recall rate

although input data has high noise rate. Thus, it can be considered that QBAM has a

superior performance in terms of noise tolerance.

From the results of simulation experiments in terms of memory capacity and noise

tolerance, it can be regarded that QBAM has a superior performance compared to

conventional methods.

4. A Robot System Architecture and Computational Intelligence

for Cognitive Development

4.1. A robot system for associative memory

We developed a human–robot communication system with a robot partner based on

associative memory. Figure 5 shows the structure of the system. It is composed of a

robot partner, Microsoft Kinect (movement, voice and gesture recognition technol-

ogy), a microphone and a server PC running Julius software. Julius is an open-source

Fig. 5. The structure of robot system.

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continuous speech recognition engine with a large vocabulary.26 Kinect extracts

RGB and distance data. Using these data as input, the server PC can detect an

object's color and shape using a k-means algorithm and a steady-state genetic al-

gorithm (SSGA), or recognize gestures using spiking neural networks (SNNs).9 The

microphone collects human voice signals for Julius. Julius connects with the server

PC by TCP/IP. The server PC also calculates the relationship between the object,

gesture and words using QBAM. Then, based on this relationship, the server PC

sends the words or behavior order to the robot partner using TCP/IP. Figure 6

shows a sample of robot behaviors.

4.2. Computational intelligence for a robot partner

Figure 7 shows example of object and gesture recognition in our system. For object

recognition, we focused on colored objects and shape recognition using SSGA based

on template matching via image processing. Furthermore, we applied a k-means

algorithm for the clustering of candidate templates in order to ¯nd several objects

simultaneously. For gesture recognition, we used OpenNI to extract a human hand

position. SNNs were applied in order to memorize the spatio-temporal patterns of

gesture, and to classify gestures; furthermore, we applied a self-organizing map

(SOM) using pulse outputs from the SNNs. SNNs are pulse-coded neural networks

Fig. 6. The example of robot partner behaviors (lower bye-bye, upper bye-bye and up & down).

(a) (b)

Fig. 7. Example of computational intelligence for recognition. (a) object. (b) gesture.

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which memorize spatial and temporal contexts. We used a simple spike response

model to reduce the computational cost. As mentioned above, we applied Julius

software for voice recognition.26 Julius works in real time, and recognition accuracy

has been shown to be over 90% in a 20,000-word reading test. We utilized color,

shape, gesture and word information as input data for associative memory.

5. Experimental Results

This section presents the experimental results of the communication between robot

partner and human using proposed model through object recognition, gesture rec-

ognition and voice recognition. We de¯ned ¯ve types of relationship for associative

memory between object, gesture and word in Table 1. If we input information to the

system, corresponding information with Table 1 will be recalled. There are three

modalities (object, gesture and word) in each ID as Table 1. Thus, input and recalled

signal show same ID. Here, each input and output are composed by bipolar pattern.

Figure 8(a) shows the input sequentially as object, gesture and word. Figures 8(b)–

8(d) show sequentially output of relationship that is associated by BAM, GBAM and

(a) (b)

(c) (d)

Fig. 8. The input ID and the result of output ID as relationships. (a) Input ID. (b) BAM. (c) GBAM.

(d) QBAM.

Table 1. Relationship between object, gesture and word.

ID Relationship

Object Gesture Word

0 No object No gesture No word1 Red circle Lower bye-bye Red circle

2 Green triangle Upper bye-bye Green triangle

3 Blue rectangle Up & down Blue rectangle

4 Blue rectangle ��� ���

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QBAM, respectively. According to Table 1, the output waveform should show the

same form as the input one except for input ID 4.

For ID 4, if there is an input as unknown (not stored) signal, the system regards

that unknown signal as no input. This is because of our de¯nition. In Fig. 8(b), due to

the quite low memory capacity, BAM cannot recall the correct output. The result of

GBAM is slightly improved from BAM, because it has a better memory capacity

than BAM. However, recall process has still some failures in GBAM. On the other

hand, in Fig. 8(d), output of QBAM show the same waveform with input one except

for input ID 4, thanks to the large memory capacity and the high recall reliability.

Figures 9–11 show the recall rate between input and output in GBAM and QBAM.

Here, in each ¯gures, the axis labels on the right-hand side represent input information,

the axis labels on the lower side represent recalled information. This relationship

follows Table 1. In Fig. 9(a), most of the recall rates do not exceed 90%. In contrast, in

Fig. 9(b), for each input, the two outputs in accordance with Table 1 indicate the

perfect recall rate. Similarly, Figs. 10 and 11 have the same tendency in results.

From the results of experiment with robot system, we regard that QBAM is

e®ective method for the associative communication system with robot partner.

(a) (b)

Fig. 9. The result of recall rate (input: object, recall: gesture and word). (a) GBAM. (b) QBAM.

(a) (b)

Fig. 10. The result of recall rate (input: gesture, recall: word and object). (a) GBAM. (b) QBAM.

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6. Conclusion

This paper has proposed aQBAM. It discussed the capability of the proposedmodel for

human–robot interaction to process color, shape, gesture and word information. We

have showed the similarity between fuzzy inference and quantum mechanics, and

proposed method satis¯ed basic postulates of quantum mechanics through mathe-

matical proofs. The simulation results showed that the proposed model has superior

abilities in terms of memory capacity and noise tolerance. In addition, the experiment

with a robot system showed the e®ectiveness of the proposed model for human–robot

interaction. From these results, we argue that the proposedmodel is the ¯rst successful

attempt to overcome several problems found in practical robot applications. In addi-

tion, if we de¯ne more relationships and di®erent information of object, gesture or

word, it will be much e®ective for communication between robot partner and humans.

As future works, we will add more relationships and develop the quantum-in-

spired multidirectional associative memory. It has more e®ectiveness for communi-

cation because it can be associated from one thing to many things at the same time.

Furthermore, we will apply complex value to fuzzy inference to represent oscillation.

The oscillation is one of the important factors in quantum mechanics. It can be

expected to improve memory capacity and noise tolerance.

Acknowledgments

The authors would like to acknowledge a scholarship provided by the University of

Malaya (Fellowship Scheme). This research is supported in part by HIR grant UM.

C/625/1/HIR/MOHE/FCSIT/10 from the University of Malaya.

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Page 22: Quantum-Inspired Bidirectional Associative Memory … · Quantum-Inspired Bidirectional Associative Memory for Human–Robot Communication Naoki Masuyama* and Chu Kiong Loo† Faculty

Naoki Masuyama graduated from Nihon University, Chiba,

Japan in 2010. He received the M.E. degree from Tokyo Metro-

politan University, Tokyo, Japan in 2012. He is currently working

toward the Ph.D. degree at University of Malaya, Kuala Lumpur,

Malaysia. His research interest is computational intelligence for

human–robot interaction.

Chu Kiong Loo obtained his Ph.D. from University Sains

Malaysia, B.Eng. (First class Hons in Mechanical Engineering)

from University Malaya.

Formerly he was a design Engineer in various industrial ¯rms

in di®erent capacities as well as he is the founder of Advanced

Robotics Lab in University of Malaya.

He has been involved in the application research of Peruss

Quantum Associative Model and Pribrams Holonomic Brain

Model in humanoid vision projects. Currently, he is the Professor of Computer

Science and Information Technology, University of Malaya, Malaysia. He has com-

pleted many funded projects by Ministry of Science in Malaysia and High Impact

Research Grant from Ministry of Higher Education, Malaysia. Loos research expe-

rience includes brain inspired quantum neural network, constructivism inspired

neural network, synergetic neural networks and humanoid research.

Naoyuki Kubota graduated from Osaka Kyoiku University in

1992, received the M.E. degree from Hokkaido University in 1994,

and received the D.E. degree from Nagoya University, Japan in

1997. He joined Osaka Institute of Technology, in 1997. He joined

the Department of Human and Arti¯cial Intelligence Systems,

Fukui University as an associate professor in 2000.

He joined the Department of Mechanical Engineering, Tokyo

Metropolitan University in 2004. He was an associate professor

from 2005 to 2012, and has been a professor since 2012 at the Department of System

Design in Tokyo Metropolitan University, Japan.

N. Masuyama, C. K. Loo & N. Kubota

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