An Iterative Incremental Learning Algorithm for Complex-Valued Hopfield Associative Memory

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1 An Iterative Incremental Learning Algorithm for Complex-Value Hopfiel Associative Memory aoki Masuyama (B) an Chu Kiong Loo Faculty of Computer Science an Information Technology, University of Malaya, Kuala Lumpur, Malaysia Abstract. This paper iscusses a complex-value Hopfiel associative memory with an iterative incremental learning algorithm. The mathematical proofs erive that the weight matrix is approximate as a weight matrix by the complex-value pseuo inverse algorithm. Furthermore, the minimum number of iterations for the learning sequence is efine with maintaining the network stability. From the result of simulation experiment in terms of memory capacity an noise tolerance, the propose moel has the superior ability than the moel with a complexvalue pseuo inverse learning algorithm. Keywors: Associative memory Complex-value moel Incremental learning 1 Introuction In the past ecaes, numerous types of Artificial Intelligence moels have been propose in the fiel of computer science in orer to realize the humanlike abilities base on the analysis an moeling of essential functions of a biological neuron an its complicate networks in a computer. It has been note that one of the interesting an challenging subjects is the imitation of memory function of the human brain. In the past ecaes, several types of artificial associative memory moels an its improvements have introuce such as Hopfiel Associative Memory (HAM) [7], an Bi-/Multi-irectional Associative Memory (BAM, MAM) [6,11]. However, the majority of moels are limite as offline an one-shot learning rules. Storkey an Valabregue [14] propose an incremental learning algorithm for HAM base on the Hebb/Anti-Hebb learning. Chartier an Boukaoum [3] also propose an analyze the moel with a self-convergent iterative learning rule base on the Hebb/anti-Hebb approach, an a nonlinear output function Dieerich an Opper [4] propose a Wirow-Hoff type learning. However, the convergence of these moels is quite slow when the memory patterns are correlate. The local iterative learning, on the other han, which is propose by Blatt an Vergini [2] can be ealt with above kin of problems. c Springer International Publishing AG 2016 A. Hirose et al. (Es.): ICOIP 2016, Part IV, LCS 9950, pp , DOI: /

2 424. Masuyama an C.K. Loo Moreover, this moel is able to efine the minimum number of learning iterations with maintaining a network stability. In regar as events in the real worl, information representation using binary or bipolar state is insufficient. Jankowski et al. [9] have introuce complex-value Hopfiel Associative Memory (CHAM) with neurons processing a complex-value iscrete activation function. Conventionally, numerous stuies that improve complex-value moels are introuce base on the improvements for real-value moels. Lee [13] applie a complex-value projection matrix to CHAM (PInvCHAM) an analyze the stability of the moel by using energy function. The learning algorithm of these moels, however, are characterize by a batch learning. Isokawa et al. [8] analyze the stability of complex-value Hopfiel moel with from a local iterative learning scheme viewpoint though the learning algorithm is Hebb learning base. In this paper, we introuce a local iterative incremental learning with CHAM base on Blatt an Vergini learning algorithm [2]. Here, we shall call the propose moel as BVCHAM. The noteworthy features of BVCHAM are escribe as follows; (i) the resultant of weight matrix in BVCHAM is approximate to the projection matrix operation. The network is able to store the patterns though the number of store patterns p is larger than the number of neurons (It is known that the memory capacity is limite as p<with a projection matrix learning moel.), an (ii) the learning algorithm is guarantee to calculate a weight matrix within a finite number of iterations with keeping a network stability. The paper is ivie as follows; Sect. 2 escribes the ynamics of BVCHAM. Section 3 presents the network stability analysis for BVCHAM. In Sect. 4, it will be presente simulation experiments of BVCHAM in terms of the memory capacity an noise tolerance comparing with PinvCHAM. Concluing remarks are presente in Sect Dynamics of a Local Iterative Learning for CHAM This section presents the ynamics of a propose moel, we shall call as BVCHAM. Let us suppose that the moel stores complex-value funamental memory vectors X p, where X p =[x p 1,xp 2,...,xp ]T, enotes the number of neurons, an p enotes the number of patterns. The components x p i are efine as follows; x p i exp [j2πn/q]q 1 n=0, i =1, 2,..., (1) where q enotes a quantization value on the complex value unit circle. Here, supposing a new pattern X l stores to the network by upating the weight matrix W (l 1) that alreay has (l 1) patterns embee. X (l) is presente n l times to the network as follows; n l W (l) = W (l 1) + ΔW (l) (2) =1

3 An Iterative Incremental Learning Algorithm for Complex-Value HAM 425 where, ΔW is efine as a following; ΔW (l) = k 1 ( x l )( ) i h i x l j h j /. (3) The local fiel h i which changes ( 1) times is performe as a following; [ ] h i = j=1 ( 1 W (l 1) ij + r=1 W (l) r ) ij x l j. (4) Similar with Eq. (4), the local fiel h j is efine. Here, the minimum number of iterations n l can be efine as a following; n l log k [/ (π/q T ) 2] (5) where, a parameter k, calle memory coefficient, is a real number that belongs to the interval 0 <k 4, an a parameter T is set between 0 T < π/q.it allows to perform in orer to achieve aligne local fiels with values of at least T [2]. q enotes a quantization value on the complex unit circle, which is escribe in a following part. The iterative weight upate is performe until satisfying the criterion as T>error ɛ, namely; ( ) T>ɛ= arg hi. (6) i=1 In summary, the process of weight upate is escribe as Algorithm 1. For the association process, the self-connections eliminate weight matrix W is utilize. On the other han, the weight matrix W which is calculate by Algorithm 1 is applie to the further incremental learning process. X l i Algorithm 1. An algorithm for a weight matrix W (l) Require: weight matrix W (l 1), funamental memory vector X l, error parameter T,memorycoefficient k Ensure: weight matrix W (l) if l =1then Initialize W as a zero matrix en if Set =1 Calculate an error ɛ using Eq. (6) while n l an ɛ>t o Calculate a weight matrix W (l) using Eqs. (2) an (3) Calculate an error ɛ using Eq. (6) Set +1 en while The activation function is a complex projection function that operates on each component of the state vector as a following; { Z exp(j2πn/q), If arg exp(j2πn/q)} <π/qan Z 0 φ(z) = (7) previous state, If Z = 0

4 426. Masuyama an C.K. Loo where, arg(α) enotes the phase angle of α which is taken to range over ( π, π). q enotes a quantization value on the complex unit circle, n takes an integer. We utilize a iscrete complex unit circle moel to etermine recalle signals. Thus, the ynamics of network is summarize as follows; h (t) = W X (t) (8) ( ) X (t+1) = φ h (t) (9) The stationary conitions for the recalle memory vector are escribe as a following; ( ) h 0 arg X p < π q. (10) 3 etwork Stability Analysis In this section, base on Blatt an Vergini algorithm [2], the conitions of network stability in BVCHAM will be iscusse, an it will be erive a criterion for the number of iterations n l to guarantee the stability of X l. Here, Dirac notation in C is utilize for simplifying the representation of proofs. Thus, Eqs. (3) an (4) can be escribe as follows; ΔW l = k 1 Xl h X l h (11) [ ] 1 h = W (l 1) + X l (12) r=1 ΔW l r here, it satisfies X l h = X l h. First of all, the changes of weight connection when a new memory vector is repeate to the network is analyze. Here, ΔW (l) (1 ) is efine as a following; r=1 ΔW (l) r = Q (l) E (l 1) X l X l E (l 1) (13) where, E (l) = I W (l) (14) here, I enotes an ientity matrix which has same imension with W (l). It assumes that Eq. (13) hols for -th iteration in orer to prove ( + 1)-th iteration. The ifference of weight connection between the -th an ( + 1)-th iteration with a memory vector X l an its local fiel h is expresse by Eqs. (12), (13) an (14), as follows; X l h =(1 a (l) Q (l) )E(l 1) X l (15)

5 An Iterative Incremental Learning Algorithm for Complex-Value HAM 427 where, a (l) = Xl E (l 1) X l. (16) Thechangein( + 1)-th iteration is expresse by Eqs. (11) an (15) asa following; ( ) 2 ΔW (l) +1 = k 1 a (l) Q (l) E (l 1) X l X l E (l 1). (17) Comparing with Eqs. (13) an (17), it can be regare as a proportional relation with the same operator. Thus, the following recurrence relations are obtaine; Q (l) 1 = 1 (18) Q (l) = Q(l) 1 + k 1 ( 1 a (l) Q (l) 1) 2. (19) From the Eqs. (13) an (14), Eq. (2) is represente as a following; E (l) = E (l 1) Q n (l) E (l 1) X l X l E (l 1) l. (20) Therefore, when a new memory vector is presente to the network, an operator which is proportional to the projector onto E (l 1) X l is ae to the weight connection. Accoring to Eq. (11), ΔW increases exponentially with the number of iterations. However, if the following conition is satisfie, the elements of ΔW remain finite, i.e.; 0 φ E (l) φ φ φ for all φ C. (21) In the first learning step, it maintains E (0) = I an Q (1) = 1. Furthermore, base on Cauchy Schwarz inequality, Eq. (21) is verifie for l = 1. Here, assumingthateq.(21) is vali for l. Then, the eigenvalues of E (l) take between 0 to 1. Therefore, the following conition is satisfie; E (l) φ 2 φ E (l) φ. (22) Let us suppose that a (l+1) =0,E (l) X l+1 = 0 is erive from Eqs. (16) an (22), then E (l+1) = E (l) is maintaine from Eq. (20). Thus, Eq. (21) is also satisfie for (l+1). In contrast, in case of a (l+1) 0, it is prove by the inuctive hypothesis 0 <a (l+1) 1. Furthermore, accoring to [2], following conitions are maintaine; 0 <a (l) Q (l) 1 a (l) Q (l) 1. (23) ). (24) ( k a (l) From the following part, it focuses the memorize pattern X μ (1 μ<l) an its local fiel h are able to be closer than other memorize patterns. The

6 428. Masuyama an C.K. Loo istance between X μ an h ecreases exponentially with the number of iterations. This is erive by the inuction process. First of all, the following conition is maintaine by the Eqs. (14) an (22); X μ W (l) X μ 2 X μ E (l) X μ. (25) Here, Eq. (20) is generalize as a following; E (l) = E (μ) l ν=μ+1 Q ν E (ν 1) X ν X ν E (ν 1) n ν. (26) Then, X μ multiplie on the left han sie, an X μ multiplie on the right han sie to Eq. (26), therefore; X μ E (l) X μ X μ E (μ) X μ. (27) For the Eq. (20) with a μ-th memory vector, X μ is multiplie on the left han sie, an X μ is multiplie on the right han sie, i.e.; X μ E (μ) X μ = X μ E (μ 1) X μ Q n (μ) ( X μ E (μ 1) X μ ) 2 μ. (28) Consiering with Eq. (16), Eq. (28) is escribe as a following; ( ) X μ E (μ) X μ = a (μ) 1 a (μ) Q n (μ) μ. (29) Furthermore, an a (μ) multiplie to Eq. (24) with a μ-th memory vector, i.e.; a (μ) ( 1 a (μ) Q (μ) n μ ) k nμ. (30) From Eqs. (25), (27), (29) an (30), the following conition is obtaine; X μ W (l) X μ 2 X μ E (l) X μ X μ E (μ) X μ = a (μ) ( 1 a (μ) Q (μ) n μ ) k nμ. (31) Accoring to the stability conition as Eq. (10), Eq. (31) is escribe as a following; X μ W (l) X μ 2 k nμ ( ) 2 π q T (32) where, q enotes a quantization value on the complex unit circle. The optimal stability can be controlle by the parameter T [5, 12]. In aition, the network stability is guarantee in case of T = 0. Furthermore, from Eq. (32), the minimum number of iterations n l is erive as Eq. (5).

7 An Iterative Incremental Learning Algorithm for Complex-Value HAM Simulation Experiment This section presents the simulation experiments comparing with a pseuo inverse learning moel (PInvCHAM) [1] an a propose moel (BVCHAM), in terms of memory capacity an noise tolerance. 4.1 Conition Table 1 shows the simulation conitions to evaluate the memory capacity an the noise tolerance. In this experiment, the number of pairs p is increase from 1 to 220 at intervals of 10 (p =1, 10, 20,...,220). oise tolerance is a significant property for the associative memory. Typically, noise is roughly ivie into two types in associative memory. One is the similarity of the store patterns, another is the store patterns contain noise itself. Due to the propose moel has the similar properties with a pseuo inverse learning moel, it is expecte that the propose moel is able to learn the correlate memory vectors without errors. In this paper, therefore, we consier about a latter type of noise which is containe in an initial input with 0 to 50 [%] by the salt & pepper noise. Furthermore, it is known that the ability of a complex-value associative memory is epenent upon the number of ivisions of a complex-value unit circle. Here, we set 4, 8 an 16 ivisions for this simulation. Table 1. Simulation conition. umber of pairs p : umber of neurons : 200 umber of ivisions q: 4,8,16 Data set configuration: Amplitue: 1.0, Phase: Ran 4.2 Result In Fig. 1, PInvCHAM maintains the high recall rate, especially R = 0.0. However, ue to the limitation of the pseuo inverse learning, the recall rate is suenly roppe uner the conition p. As shown in Fig. 2, ue to the weight matrix of BVCHAM can be approximate as the weight matrix by a pseuo inverse learning, the recall rate of BVCHAM shows the similar or superior results (especially in Fig. 2(a)) than PInvCHAM. Here, it focuses on the results with R =0.0 in Fig. 2. Since the weight matrix of BVCHAM is not equal to the ientity matrix at p =, BVCHAM is able to maintain the high recall rate than PInvCHAM even if the conition is uner p. From the above results, it is regare that BVCHAM has the following noteworthy avantages; although a propose learning algorithm is incremental, the ability can be comparable to a batch learning as a pseuo inverse learning algorithm, an it is able to overcome the limitation of a pseuo inverse learning algorithm that is characterize by p<.

8 430. Masuyama an C.K. Loo (a) q =4 (b)q =8 (c) q =16 Fig. 1. Results of recall ratio for a pseuo inverse learning moel. (a) q =4 (b)q =8 (c) q =16 Fig. 2. Results of recall ratio for a BV incremental learning moel. 5 Conclusion This paper introuce a local iterative incremental learning algorithm for a complex-value Hopfiel associative memory. Furthermore, we presente the network stability analysis which erive the weight matrix of BVCHAM is approximate as a weight matrix from a complex-value pseuo inverse learning algorithm, an the minimum number of iterations with maintaining a network convergence. From the result of simulation experiment in terms of memory capacity an noise tolerance, BVCHAM has the superior ability than the moels with the Hebb learning an a complex-value pseuo inverse learning algorithm. oteworthy, unlike the moel with a pseuo inverse learning algorithm, the propose moel is able to maintain the high recall rate even if the number of store patterns is larger than the number of neurons. As a future work, we will exten the propose incremental learning algorithm to hetero-association moels, such as bi/multi-irectional associative memory moels [10]. Acknowlegments. The authors woul like to acknowlege a scholarship provie by the University of Malaya (Fellowship Scheme). This research is supporte by High Impact Research UM.C/625/1/HIR/MOHE/FCSIT/10 from University of Malaya.

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