Elsevier

Neurocomputing

Volume 30, Issues 1–4, January 2000, Pages 249-272
Neurocomputing

Co-adaptation of self-organizing maps by evolution and learning

https://doi.org/10.1016/S0925-2312(99)00129-0Get rights and content

Abstract

This paper proposes two co-adaptation schemes of self-organizing maps that incorporate the Kohonen's learning into the GA evolution in an attempt to find an optimal vector quantization codebook of images. The Kohonen's learning rule used for vector quantization of images is sensitive to the choice of its initial parameters and the resultant codebook does not guarantee a minimum distortion. To tackle these problems, we co-adapt the codebooks by evolution and learning in a way that the evolution performs the global search and makes inter-codebook adjustments by altering the codebook structures while the learning performs the local search and makes intra-codebook adjustments by making each codebook's distortion small. Two kinds of co-adaptation schemes such as Lamarckian and Baldwin co-adaptation are considered in our work. Simulation results show that the evolution guided by a local learning provides the fast convergence, the co-adapted codebook produces better reconstruction image quality than the non-learned equivalent, and Lamarckian co-adaptation turns out more appropriate for the VQ problem.

Introduction

Vector quantization (VQ) [6] is a simple and effective data compression technique that uses vectors instead of scalars to achieve low bit-rate and better image reconstruction quality [24]. Mathematically, a vector quantizer Q of dimension k and size N is a mapping from an arbitrary vector X={x1,x2,…,xk}∈Rk into a finite set C such that Q:Rk→C, where C is called a codebook that consists of N codevectors C=(Y1,Y2,…,YN). The vector quantizer is completely specified by determining N reproduction codevectors C and their corresponding non-overlapping partitions R=(R1,R2,…,Rn) called Voronoi regions. A Voronoi region Ri is defined asRi={X∈Rk:∥X−Yi∥≤∥X−Yji≠j}and represents a subset of vectors of Rk, where the vectors are nearest to the codevector Yi in the meaning of L2 norm ∥·∥.

A design problem of a good codebook is an important subject for an optimal vector quantization. Among the commonly used VQ design methods, the Linde–Buzo–Gray algorithm [18] using a clustering approach has been the most well-known one. As an alternative VQ design method, an unsupervised neural network called the self-organizing map (SOM) [15] that is adapted by Kohonen's learning rule has been widely applied to codebook design for vector quantization [12], [13], [23]. These algorithms are considered as the suboptimal ones at best because they are based on the local gradient descent method for which a prescribed distortion measure is decreased monotonically by updating the codebook entries iteratively. Furthermore, due to the highly non-linear nature of the VQ problem, the distortion function is non-convex and may exhibit many local minima. Therefore, the resultant codebook does not guarantee the minimum distortion and its reconstruction performance is sensitive to the choice of initial parameters.

Many researchers have applied the evolutionary approach to the codebook design for VQ. Pan et al. [25] have proposed the evolution of the VQ codebook by the genetic algorithm (GA) in order to tackle the above problems. However, they did not use the mutation in genetic operation and their codebook size was too small to validate the practical usage. Kim and Ahn [14] have applied the GA to design a good codebook robust to the bit error for wireless communication. McInerney and Dhawan [21] proposed a hybrid algorithm of genetic methods with Kohonen learning that avoided a topologically incorrect mapping between input and output spaces due to a poor choice of parameters. However, the chromosome in their approach consisted of both the codevectors and the learning parameters. So, it is not easy to optimize both codevectors and the learning parameters simultaneously. Polani [27] introduced three different fitness functions adequate to optimize the SOM's topology such as the quantization errror, the Hebbian measure, and the hybrid of both. They argued that the hybrid organization measure is a significant discrimination of organization quality for SOMs in the two-dimensional case and is a promising candidate for a GA search of SOM topology in the higher dimensional case. However, his approach used the only GA to get the codebook for VQ. Polani and Uthmann [28] demonstrated that a coupling of the GA and the Kohonen Featuer Map paradigm led to reasonable results regarding the capability of adaptation to a given event space. In particular, Kohonen networks proved to be promising candidates for phenotype–genotype mapping by a fairly simple transcription rule due to their highly uniform structure.

Recently, the idea of combining evolution and learning based on concepts inspired from biological systems has attracted much attention. Evolution changes the genotype of an organism at the population level while learning changes the behavior of an organism at the individual level. Change by evolution is made by reproducing an organism selectively and performing genetic operations (recombination and/or mutation) in order to maintain inter-individual variability. The change due to evolution is generationally cumulative in a manner that some changes at a particular generation is superimposed upon other changes in the previous generations. Change by learning is made by the interaction of an organism with a specific environment during its lifetime and incorporating aspects of the environment through its experience in its internal structure. The change due to learning is individually cumulative in a manner that some changes in an individual at a particular time of its lifetime are influenced by other changes at preceding times.

Although evolution and learning are two distinct types of changes that occur in two distinct kinds of entities (population and individual organism, respectively), they may influence each other. The influence of evolution on learning can be easily understood by the fact that an evolutionary change leaves its trace in the genotype and an individual's genome partially specifies the individual's phenotypic traits and it constrains how the individual will behave in the future and what it will learn. The effect of evolution on learning has been found in many simulations that apply genetic algorithms to the population of neural networks. Evolution progressively selects for networks that incorporate a predisposition to learn some specific task. The predisposition to learn some specific task can be incorporated into neural networks in various ways. For example, evolution may select initial weight vectors [4] or network architectures [22] that cause better learning.

There are two hypotheses on how learning influences evolution. One is the Lamarckian hypothesis [16] that phenotypic traits acquired by an organism during its lifetime is transcripted into the heritable genotype and can be passed on to the organism's offspring directly. Ackley and Littman [1] showed that Lamarckian evolution could be both easy to implement and potentially far more effective for optimization problems due to its fast convergence. Another is the Baldwin effect [3] that acquired characteristics are not inherited to its offspring directly but adaptive learning can guide the course of evolution indirectly in a way that learning alters the shape of search space and thereby provides good evolutionary paths towards sets of co-adapted individuals. Hinton and Nowlan [10] demonstrated that the Baldwin effect allowed the learning organisms to evolve faster than their non-learning equivalents even though the characteristics acquired by the phenotypes are not directly communicated to their genotypes. Parisi and Nolfi [26] also indicated that performing a learning task that was indirectly related to an organism's fitness could still be used to guide evolution. They used a crude estimate of the direction towards the local minimum as indirect learning. According to the result of their experiment, coarse approximation to the gradient can provide enough directionality to allow learning to successfully guide evolution.

Fig. 1 illustrates two distinct types of interactions between evolution and learning. According to the viewpoint of modern biology, Lamarckian hypothesis is not accepted any more. However, from the viewpoint of computer scientists, computational version of the Lamarckian model can be easily implemented in the artificial simulated system. So, we adopt both co-adaptive mechanisms incorporating learning into evolution for an optimal VQ codebook design. The adopted co-adaptation method will adjust the codevector entries in the SOM by evolving many codebooks with GAs and learning each individual codebook with Kohonen's learning rule.

This paper is organized as follows. A VQ codebook design based on Kohonen's learning rule is presented in Section 2. An evolutionary VQ codebook design based on genetic algorithms is explained in Section 3. A new codebook design by co-adaptation that incorporates learning into evolution is proposed in Section 4. Simulation results are performed to demonstrate faster convergence and better image quality in terms of the number of generations i and the root mean square distortion error, respectively, in Section 5. Finally, a conclusion is drawn.

Section snippets

Codebook design by Kohonen's learning rule

The Kohonen's learning rule is an iterative procedure for training a sheet-like artificial neural network called SOM. The learning procedure is unsupervised or self-organized and is used for tuning the nodes to input vector distributions. The training of the SOM is initialized by assigning small random values to the weight vectors C={Y1,Y2,…,YN} of the units in the network. Each iteration in the learning process consists of three steps: the presentation of an input vector to the network, the

Codebook design by evolution

GAs [11] are the population-based searching and optimization techniques that encode a potential solution to a specific problem on a simple chromosome-like data structure and apply the genetic operators (selection, recombination, and mutation) to the chromosomes in order to achieve a better solution. It is believed that the GAs can provide a further possibility of obtaining the global optimal solution because their trials are allocated in an exponentially differentiated way to a large number of

Codebook design by co-adaptation with evolution and learning

Before introducing a co-adaptive algorithm for an optimal codebook design, we explain why learning does have a beneficial effect on evolution by a simple figure. Fig. 4 illustrates a fitness curve of all possible weight vectors in the SOM neural network. Consider two individuals a and b that have two different genomes. Therefore, they are located in two different locations in weight space. Assume that the genomes of two individuals a and b have the same fitness. When evolution is performed

Simulation results and discussion

The proposed co-adaptive codebook design techniques are applied to vector quantization of images and their convergence characteristics and visual performances are compared among different combinations of evolution and learning in terms of average distortion measure against the number of GA generations (or Kohonen's learning iterations). Simulations for obtaining an optimal codebook are performed using an image “Lena” of 512×512 pixels and 8 bits per pixel. The simulation conditions are given as

Conclusion

This paper proposes a new codebook design method that utilizes biologically inspired concepts such as the Lamarckian and Baldwin co-adaptations. The conventional codebook design using a self-organizing map and Kohonen's learning rule has the disadvantages that the convergence characteristic and the resultant codebook performance are too sensitive to the choice of initial weights and learning parameters. In particular, it is almost impossible to determine the initial weights and initial learning

Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable comments, which improved the presentation and contents of this paper considerably.

Daijin Kim received the B.S. degree in electronic enginerring from Yonsei Univerisity, Seoul, Korea, in 1981, and the M.S. degree in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Taejon, in 1984. In 1991, he received the Ph.D. degree in electrical and computer engineering from Syracuse University, Syracuse, NY. He is currently an Associate Professor in the Department of Computer Engineering at DongA University, Pusan, Korea. His research interests

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    Daijin Kim received the B.S. degree in electronic enginerring from Yonsei Univerisity, Seoul, Korea, in 1981, and the M.S. degree in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Taejon, in 1984. In 1991, he received the Ph.D. degree in electrical and computer engineering from Syracuse University, Syracuse, NY. He is currently an Associate Professor in the Department of Computer Engineering at DongA University, Pusan, Korea. His research interests include intelligent systems, computational intelligence, and hardware implementation of intelligent systems.

    Sunha Ahn received the B.S. and M.S degrees in computer engineering from DongA University, Pusan, Korea, in 1996 and 1998, respectively, and is currently working towards the Ph.D. degree in computer engineering from DongA University. Her research interests include image processing, computational intelligence, and information retrieval.

    Dae-Seong Kang received the B.S. degree in electronic engineering from Kyungpook National University, Taegu, Korea, in 1984, and the M.S. and Ph.D. degree in electrical engineering from Texas A&M University, College Station, TX, in 1991 and 1994, respectively. From 1984 to 1989 he was with Agency Defence Development, where he worked in communication device development. From 1994 to 1995 he was with electronics and Telecommunications Research Institute, Taejeon, Korea, where he was involved in the development of the setellite broadcast system. Since 1995, he has been with the department of electronic engineering, DongA University, Pusan, Korea. His research interest include image processing and compression, computer vision, and multimedia communication.

    This paper was supported by the grant of KOSEF(981-0920-104-2) of Republic of Korea.

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