2007 IEEE International Conference on Granular Computing
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Granular Computing in the information transformation of Patten recognition Hong Hu Zhongzhi Shi
Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Science Beijing 100080, China E-mail:
[email protected]
such as classes, clusters, subsets, groups and intervals to build an efficient computational model for complex applications with huge amounts of data, information and knowledge[2]. Just as the scholars summarized in the in the IEEE-GrC’2006 conference “though the label is relatively recent, the basic notions and principles of granular computing, though under different names, have appeared in many related fields, such as information hiding in programming, granularity in artificial intelligence, divide and conquer in theoretical computer science, interval computing, cluster analysis, fuzzy and rough set theories, neutrosophic computing, quotient space theory, belief functions, machine learning, databases, and many others[2]. The above definition of Granular Computing (GrC) is too augmental and the subjects about classes, clusters, subsets, groups and intervals have already studied by artificial intelligence or mathematics for a long time. What is really new point for GrC ? We think that the new or main point of the Granular Computing (GrC) lays on the original insight of Granular Computing (GrC) proposed by Zadeh that there are three basic concepts that underlie human cognition: granulation, organization and causation, and informally, granulation involves decomposition of whole into parts; organization involves integration of parts into whole; and causation involves association of causes with effects. Granulation of an object A leads to a collection of granules of A, with a granule being a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality[1]. In this original insight of Granular
Abstract In the past decade, many papers about granular computing(GrC) have been published, but the key points about granular computing(GrC) are still unclear. In this paper, we try to find the key points of GrC in the information transformation of the pattern recognition. The information similarity is the main point in the original insight of Granular Computing (GrC) proposed by Zadeh(1997[1]). Many GrC researches are based on equivalence relation or more generally tolerance relation, equivalence relation or tolerance relation can be described by some distance functions and GrC can be geometrically defined in a framework of multiscale covering, at other hand, the information transformation in the pattern recognition can be abstracted as a topological transformation in a feature information space, so topological theory can be used to study GrC. The key points of GrC are (1) there are two granular computing approaches to change a high dimensional complex distribution domain to a low dimensional and simple domain,(2)these two kind approaches can be used in turn if feature vector itself can be arranged in a granular way . Key words : GrC, information Trans. , cognition.
1. Introduction In the IEEE-GrC’2006 conference of information about the Granular Computing (GrC), the outline of Granular Computing is defined as a general computation theory for effectively using granules 1
This work is supported by the National Natural Science Foundation of China (No. 60435010, 60675010), 863 National High-Tech Program (No.2006AA01Z128), National Basic Research Priorities Program (No.2007CB311004) and the Nature Science Foundation of Beijing (No. 4052025).
0-7695-3032-X/07 $25.00 © 2007 IEEE DOI 10.1109/GrC.2007.42
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Computing (GrC), Zadel pointed out three important points about GrC : (1) the Granular Computing (GrC) is a main character of human cognition, (2) so called Granular Computing (GrC) is based on indistinguishability, similarity, proximity or functionality, (3) there is a close relation among granulation, organization and causation. Based on these points, we think that it is necessary to find some key points of GrC in human cognition. There are two levels of Granular Computing (GrC) research– perception and knowledge level. In this paper we focus in the perception level. Indistinguishability, similarity and proximity can be described by equivalence relation or tolerance relation, and these relations can be described by some kind distance functions. Many papers of Granular Computing (GrC) study classes and clusters, [45][17-18]. Zhang Ling and Zhang Bo [17-20] try to use the quotient space theory to study indistinguishability and similarity. Yao,Y.Y [23] extends the equivalent class to rough approximation set . The quotient space structure described by
based on tolerance relation can be used to describe domain stricture. The domain structure of a sample space represents indistinguishability, similarity and proximity of examples. From geometric point of view, the domain structure represents the shape features of this domain. （2） Information transformation used in pattern recognition should be taken place in a granular system based on tolerance relation. This is the main point we shall discuss in this paper. Now we try to use fuzzy logical formula based on distance function to define granular systems. There (d2) are three distance axioms: (d1) d(α, α)=0; d(α,β)= d(β, α); (d3) d(α,β)+ d(β,c) ≥ d(α,c). dis(x,y)r （ strong
equivalence relation is used to probe the structure of granules such as classes, clusters, subsets, groups etc. In a more general way, T.Y.Lin [7][8] and Yao, Y.Y [24][25] use binary relations to study indistinguishability and similarity, the geometric concepts- partions, covering and topology and neighborhood can be described by binary relations in the algebra. At other hand, information transformation which is neglected by most researches of GrC can be abstracted as a general meaning topological transformation, so in this paper topological theory is used to study GrC, and we try to find some important points of GrC in the pattern recognition. 2. Granular system based on tolerance Relation The difference of a granular system based equivalence relation and granular system based
on
tolerance relation is that an equivalence relation will
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r_cut set）or s p r(α, c|dis,d,ω) ＝ sp(α, c|dis,d,ω)≥r （r-cut set）. spr(x,c|dis,d,ω) define a open convex region and
divide a space into nonoverlapping covering and a tolerance relation will create overlapping covering of this space.
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denoted as a granule， and s p r(x,c|dis,d,ω) define a closed convex region. Definition 2 (leveled granular system based on tolerance relation Gsys) the granular system based on tolerance relation(granular system for short) is a open set system（point to point distance is used）or class system of open sets (set to set distance is used. In this paper, only point to point distance is used. It has
Two main purposes to build a granular system based on tolerance relation. （1） Granular systems based on tolerance relation can be viewed as a topological structure built by topological bases on the topological space（X,г）induced from a metric space (X,dis）by the metric dis. Granular systems
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countable infinite or finite levels. The radiuses of granules are decreased and trend to zero when the level trends to infinite. The centers of granules will distribute in a so called center grid, usually the center grid is discrete, but it can also be a continuous set(The formal description is omitted for the sake of pages). Two kind layered computing can be take place over a granular system. For every granule, a new feature vector denoted as adjoint feature vector can be computed (Fig.1(a), (Fig.1(b)), in the simplest case, for example, an adjoint feature vector can be the centriod of a granule. 3. Granular Environment
Information
Sampling
level of Gsys, if X has n dimensions, the center grid of level l over X, i.e. Gcl(X) can be viewed as a n dimensional matrix, we denote such kind matrix as sampled information matrix sGl(X) of level l and its corresponding granular system built on it as sensordistribution granular system(sdGsys). 4. Information transformation under Granular assumption by granular systems Not losing generality, in the processing of pattern recognition, it is necessary to find borderlines of example distribution among classes for classifying or clustering. There are two kind similarity among examples-static similarity and dynamical similarity. If elements of classes are distributed in standard convex regions, we can use some kind distance function to describe classes’ distribution domains. In this case, similarity between two objects can be described by distance functions; we denote such kind similarity as static similarity. The dynamical similarity is different from static similarity, if one object O1 continuously changes to another object O2 , e.g. a tadpole continuously grow up to a frog, then O1 and O2 are dynamical similar, i.e. if all elements in a class A are dynamical similar, the distribution domain of A is a connected domain. Dynamical similarity will cause distribution domain become very complicate and have a nonlinear borderline. As to change complex borderlines to simple ones and reducing the dimension of feature space are still the most effective approaches in pattern recognition, we use the topology transformation to analyze feature changing and dimension-reducing. If F:Rn→Rk is a continuous mapping from Rn to k R , as we know F can transform any domain in Rn to a domain Rk by arbitrary stretching and squeezing. A connected domain in Rn will never be mapped to two separate domains by a continuous mapping. If F is not a one-to-one mapping, two different points in Rn may be mapped to same point in Rk, so two separated domains in Rn may be mapped to a connected domain in Rk. But if F is one-to-one, such kind thing will not occur. The famous Clasification theorem in the topology theory tells us that any closed surface is homeomorphic either to the sphere, or to the sphere with a finite number of handles added, or to the sphere with a finite number of discs removed and replaced by Mǒbius strips[11]. Unfortunately the case is not so simple when the surface’s dimension is greater than 2, we have no knowledge about the classification of surfaces in high dimensional space. For most pattern recognition task, a continuous mapping is versatile
from
3.1. Sensor-distribution granular system (sdGsys) Our universe is a complex, chaotic dynamical (a) nested layered computing
(b) unnested layered computing
Fig.1. Two kind layered computing over granular system
system. As we know information displays in a fractal way in a complex, chaotic dynamical system, i.e. when we zoom in, infinite small detailed structure of nature will be continuously explored before us, when we zoom out, infinite large structure of nature will be continuously displayed before us. The information of nature is continuous and incomputable under the meaning of the classical Turing machine. From the engineer point of view or from computable point of view, the information from nature should be sampled approximately, and only countable or finite sensors can be settled in our environment by us. The sensors distributed in our environment should be distributed in a granular way. If Gsys is a granular system on the environment space X, we can settle sensors on all centers of granules of Gsys, a sensor in the level l is denoted as Sl(c), where c is a center of a level l granule G(coeGl), the focus of Sl(c) is the whole domain represented by G(coeGl), the radius of a granule determines the focus scale of the sensor Sl(c). In this way, the complex, fractal and continuous information of our chaotic environment can be approximately sampled in a multiscale way with an arbitrary precision. If the scale of the environment space X is finite, we can suppose there are only finite granules in every
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enough to transform complex nonlinear borderlines or surfaces of distribution domains to rings or spheres, except for the distribution domains of different classes are connected together in some kind knots[Fig.3][12]. For knots kind structure, we should use several continuous functions combined with several inequalities to unlock knot-kind structure. In this thesis, we discuss a more generalized case that a continuous mapping which keeps two distribution domain of two classes separating (see the definition 8). Definition 3 A continuous mapping F: Rn→Rk , k≤n is defined as “ε separating-holding mapping” on a domain set Sr={Bi}, if for disS(Bi,Bj)≥ε>0, we have disS(F(Bi),F(Bj))≥ε>0 also, where ε is a positive constant, F(Bi) and F(Bj) are images of Bi and Bj on Rk separately. If kP(E), must distribute in a proximal way in our environment, so the matrix sGcl(X) from sampling image granular system Gsys(sGcl(X)) must be a good matrix for granular information processing. Theorem 1. By using an unnested layered computing, every ε separating-holding, region unifying and reduction mapping F(x1,…,xn)=(F1(x1,…,xn),…,Fm(x1,…,xn)) can be computed with arbitrary precision upon a sampling image granular system Gsys(sGcl(X)) or upon a sensed feature vector space granular system Gsys(SV). For more two kinds layered computing can be used in turn.(the proof is omitted for the sake of pages) 5. Conclusion The dynamical similarity will cause the distribution domain to be a connected complex domain. To change a high dimensional complex distribution domain to a low dimensional and simple domain is the task of information transformation in the pattern recognition. In fact, dynamical similarity can be viewed as a topological maping on the center grid of level l over X, i.e. Gcl(X), and ε separating-holding ,reduction and region unifying mapping tries to transfer a dynamical similarity to a static similarity. Much more details about these two mappings are unknown. So many detailed works should be done in future.
every variable xi in a region unifying mapping. In a layered computing in function structure, if variables in every the clique variable set Sq,i={xp1,…,xpq} are just distribute in a standard convex region of sGcl(X) which can be described by a set of simple fuzzy logical formulas based on distance function(a clique variable set Sq,i={xp1,…,xpq} which is just a standard convex region is denoted as granular clique variable set ), then it is easy to built a granular system to compute
References Omitted for the sake of pages.
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