M. Cottrell, J.-C. Fort, G. Pages, "Theoretical aspects of the SOM algorithm", Neurocomputing, 21, pp. 119-138, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....oriented) VQ schemes. 4.1.1 The Serf Organizing Map The main feature of the SOM is the topological structure in the index set A and the neighborhood learning based on it to achieve a topographic mapping. The treatment of the simple adaptation process is mathematically very difficult [31]. Most results have only been established for the one dimensional case. Thereby, mathematical question include the topics of 1. convergence and ordering, 2. topology preservation and 3. probability density matching and magnification. Convergence and ordering: For continuous inputs Ritter and ....

....of A are called neurons. more general, parameter settings are studied by Cottrell, Flanagan, Fort, Pag6 and colleagues, verifying the almost sure convergence in dependence of the concrete choice of the neighborhood shape and range, learning rate etc. A review of the results can be found in [31]. Meta stable states during SOM leaming may occur for certain configurations (non vanishing learning rate) 50] Sufficient conditions for convergence are given in [48] Lebesque continuous inputs are studied in [49] discrete inputs distributions in [92] For the higher dimensional cases results ....

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M. Cottrell, J. C. Fort, and G. Pages. Theoretical aspects of the som algorithm. Neurocomputing, 21(1):119 138, 1998.

....Specifically, its main property is that it conserves quite consistently the original topological and metric relationships of the items. However, despite the long history and widespread use of the method [3] its theoretical properties are still not fully understood. According to a recent review [4], only the particular SaM case for data and grid in dimension 1 has been fully worked out. Informally, one can say that the SOM algorithm usually works well, but we do not thoroughly know why . The main theoretical approach towards an understanding of the SaM algorithm in general has been ....

M. Cottrell, J.C. Fort, G. Pages, Theoretical aspects of the SOM algorithm, Neurocomputing 21 (1998) 119-138.

.... loc (f0; 1g n 2 ; V ) 2) 8u 2 R; H(u) 8 : 1; u 0 0; u 0 is the Heaviside step function; 3) d lm = kw l vmk with 8 2 [0; 1] vm = 1 ) v wm ; 4) and t is a learning factor such that P t t = 1 and P t 2 t 1, which is a necessary (but not sucient) condition (Cottrell 1998) to allow the almost sure convergence of the rule (1) toward a local minimum of the following energy function. Let the na ve energy function which the rule (1) is supposed to minimize be: E V (w) 1 2 n X i=1 Z V i (w; v) v w i ) 2 P (v)dv (5) where V is the union of the Vorono ....

M. Cottrell, J.C. Fort, G. Pages, Theoretical aspects of the SOM algorithm, In Neurocomputing, 21, pp. 119-138, Elsevier, 1998.

.... fact that there is a relation to clustering and visualization techniques is also well known, see e.g. 1] 10] 15] 4] and [24] Theoretical analysis of SOM concentrates on issues within the method (e.g. convergence) rather than commenting on how and for what SOM should actually be used (see [7] for a survey of results) However, there is also a considerable amount of criticism formulated both in terms of empirical and theoretical comparison. In [1] as well as [30] SOM is compared to various clustering algorithms on artificial data. In [2] SOM is compared to principal component analysis ....

Cottrell M., Fort J.C., Pages G.: Theoretical aspects of the SOM algorithm, Neurocomputing, (21)1-3, pp.119-138, 1998.

....centers of clusters satisfying the vector quantization criterion: E = min M # j=1 x j w b(x j ) 4) where we seek to minimize the sum squared distance E of all input patterns, x j ,j =1. M, to the respective best matching units with weight vectors w b(x j ) Furthermore [2] relates the point density function, p(w) of the weight vectors with the point density function, p(x) of the sampled underlying distribution in the V I . Density function p(w) is a better approximation of the underlying p(x) than the approximation p(x) d d 2 . Here d is the dimension of the ....

M. Cottrell. Theoretical aspects of the SOM algorithm. In Proc. of Workshop on SelfOrganizing Maps, pages 246--267. Helsinki University of Technology, 1997.

....we split the TKM algorithm in two. In the rst stage the data is Voronoi partitioned among the units with the network activity function. In the second stage the new weights given the partitioning are computed. While proving convergence for any SOM model is very diOEcult, possibly impossible [3, 2], if the TKM converges the weights have to satisfy the criteria we dene here. We have a set S = fX 1 ; XN g of discrete sequences and a map V . Last sample of each sequence X j 2 S is x j (n j ) where n j is the length of the sequence X j 2 S. In a steady state the TKM weights have to be in ....

Marie Cottrell. Theoretical aspects of the som algorithm. In Proceedings of WSOM'97, Workshop on Self-Organizing Maps, Espoo, Finland, June 4-6, pages 246267. Helsinki University of Technology, Neural Networks Research Centre, Espoo, Finland, 1997.

....the brain, but also an for adaptive nonlinear principal curve and manifold analysis [13] and feature mapping. The SOM training process is paradigmatic for many self organizing processes studied in neural systems. However, despite its algorithmic simplicity it is difficult to analyze mathematically [3, 2]. Nevertheless or perhaps even just because of that it provides a model system for studying different approaches to quantify organization in a self organizing neural network. How can order parameters be selected for SOM First, one has to make clear what one considers to be an order parameter, ....

M. Cottrell, J. C. Fort, and G. Pag`es. Theoretical aspects of the som algorithm. In WSOM '97: Workshop on Self-Organizing Maps, Espoo, Finland, June 1997.

.... is also well known, see e.g. Balakrishnan et al. 94] Flexer 97] Kohonen 97] Bishop et al. 98] and [Schwenker et al. 98] Theoretical analysis of SOM concentrates on issues within the method (e.g. convergence) rather than commenting on how and for what SOM should actually be used (see [Cottrell et al. 98] for a survey of results) However, there is also a considerable amount of criticism formulated both in terms of empirical and theoretical comparison. Balakrishnan et al. 94] as well as [Waller et al. 98] compare SOM to various clustering algorithms on artificial data. Bezdek Nikhil 95] ....

Cottrell M., Fort J.C., Pages G.: Theoretical aspects of the SOM algorithm, Neurocomputing, (21)1-3, pp.119-138, 1998.

.... of clusters satisfying the vector quantization criterion: E = minf M X j=1 jjx j Gamma w b(x j ) jjg; 4) where we seek to minimize the sum squared distance E of all input patterns, x j ; j = 1 : M , to the respective best matching units with weight vectors w b(x j ) Furthermore [2] relates the point density function, p(w) of the weight vectors with the point density function, p(x) of the sampled underlying distribution in the V I . Density function p(w) is a better approximation of the underlying p(x) than the approximation p(x) d d 2 . Here d is the dimension of the ....

M. Cottrell. Theoretical aspects of the SOM algorithm. In Proc. of Workshop on SelfOrganizing Maps, pages 246--267. Helsinki University of Technology, 1997.

.... is also well known, see e.g. Balakrishnan et al. 94] Flexer 97] Kohonen 97] Bishop et al. 98] and [Schwenker et al. 98] Theoretical analysis of SOM concentrates on issues within the method (e.g. convergence) rather than commenting on how and for what SOM should actually be used (see [Cottrell et al. 98] for a survey of results) However, there is also a considerable amount of criticism formulated both in terms of empirical and theoretical comparison. Balakrishnan et al. 94] as well as [Waller et al. 98] compare SOM to various clustering algorithms on artificial data. Bezdek Nikhil 95] ....

Cottrell M., Fort J.C., Pages G.: Theoretical aspects of the SOM algorithm, Neurocomputing, (21)1-3, pp.119-138, 1998.

.... centers of clusters satisfying the vector quantization criterion: E = minf M X j=1 jjx j Gamma w b(x j ) jjg; 4) where we seek to minimize the sum squared distance E of all input patterns, x j ; j = 1 : M , to the respective best matching units with weight vectors w b(x j ) Furthermore [4] relates the point density function, p(w) of the weight vectors with the point density function, p(x) of the sampled underlying distribution in the V I . Density function p(w) is a better approximation of the underlying p(x) than the approximation p(x) d d 2 . Here d is the dimension of the V ....

M. Cottrell. Theoretical aspects of the SOM algorithm. In Proc. of Workshop on Self-Organizing Maps, pages 246267. Helsinki University of Technology, 1997.

.... of clusters satisfying the vector quantization criterion: E = minf M X j=1 jjx j Gamma w b(x j ) jjg; 4) where we seek to minimize the sum squared distance E of all input patterns, x j ; j = 1 : M , to the respective best matching units with weight vectors w b(x j ) Furthermore [4] relates the point density function, p(w) of the weight vectors with the point density function, p(x) of the sampled underlying distribution in the V I . Density function p(w) is a better approximation of the underlying p(x) than the approximation p(x) d d 2 . Here d is the dimension of the V ....

M. Cottrell. Theoretical aspects of the SOM algorithm. In Proc. of Workshop on Self-Organizing Maps, pages 246--267. Helsinki University of Technology, 1997.

.... ascribed to its clarity and practicality: easy to write down, to simulate, to understand (at least at a basic level) and with many important practical applications (see [2] for a list of thousands of papers on self organizing maps) On the other hand, there are many theoretical issues (see e.g. [3] for a recent summary) and implications that have puzzled and probably also annoyed researchers. Quantifying topology preservation. What is a good solution When would you say that a map is well organized Many different measures have been proposed (see e.g. 4,5] but in most cases the ....

....algorithm to compute the quality of the map is much more complicated than the learning rule itself. Convergence proofs. Can you proof that the algorithm converges to such a good solution Much work has been done to proof convergence in special cases (most notably one dimensional maps, see e.g. [3] and references herein) No general proofs exist. Energy functions. Can you write down an energy function, such that the learning rule corresponds to some kind of gradient descent What is the learning rule in fact minimizing For a finite set of training patterns, it is possible to come up with ....

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M. Cottrell, J. Fort, and Pag`es. Theoretical aspects of the SOM algorithm. Neurocomputing, 21:119--138, 1998.

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M. Cottrell, J.-C. Fort, G. Pages, "Theoretical aspects of the SOM algorithm", Neurocomputing, 21, pp. 119-138, 1998.

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Cottrell, M., Fort, J.-C., & Page s, G. (1998). Theoretical aspects of the SOM algorithm. Neurocomputing, 21, 119--138.

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Cottrell M., Fort J. C., Pags G., Theoretical aspects of the SOM algorithm, Neurocomputing, 21, p. 119-138, 1998.

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M. Cottrell, J. C. Fort, G. Pags , "Theoretical aspects of the SOM algorithm", Neurocomputing, 21, p. 119-138, 1998.

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Cottrell M., Fort J.C. & Pages, Theoretical Aspects of the SOM Algorithm, Neurocomputing, 21, 1998, p. 119-138.

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M. Cottrell, "Theoretical aspects of the SOM algorithm," Neurocomputing, vol. 21, pp. 119--138, 1998.

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M. Cottrell, J.-C. Fort, and G. Pages. Theoretical aspects of the SOM algorithm. Neurocomputing, 21(1--3):119--138, 1998.

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Cottrell, M., Fort, J. C., and Pags, G. (1998b). Theoretical aspects of the SOM algorithm. Neurocomputing, 21(1--3):119--138.

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M. Cottrell, J.C. Fort, and G. Paget. Theoretical aspects of the SOM algorithm. Neurocomputing 21:119--138, 1998.

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M. Cottrell, J.C. Fort, and G. Pages. Theoretical aspects of the SOM algorithm. Neurocomputing, 21(1):119-- 138, 1998.

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Cottrell M., Fort J.C., and Pages G.: Theoretical aspects of the SOM algorithm, Neurocomputing, 21, p119-138 (1998)

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Marie Cottrell. Theoretical aspects of the som algorithm. In Pr oceedings of WSOM'97, Workshop on Self-Organizing Maps, Espoo, Finland, June 4-6, pages 246267. Helsinki UniversityofT echnology, Neural Netw orks Research Centre, Espoo, Finland, 1997.

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