T. K. Kohonen, J. A. Kangas, and J. T. Laaksonen. Variants of self-organizing maps. Neural Networks, IEEE Transactions on, 1(1):93 -- 99, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in [33] a SOM with million units was trained with 6.8 million 500 dimensional data vectors. If desired, some vector quantization algorithm, e.g. means, can be used instead of SOM in creating the first abstraction level. Other possibilities include the following. Minimum spanning tree SOM [19], neural gas [20] growing cell structures [21] and competing SOM s [22] are examples of algorithms where the neighborhood relations are much more flexible and or the low dimensional output grid has been discarded. Their visualization is much less straightforward than that of the SOM. In ....

J. A. Kangas, T. K. Kohonen, and J. T. Laaksonen, "Variants of selforganizing maps," IEEE Trans. Neural Networks, vol. 1, pp. 93--99, Mar. 1990.

....TIME Magazine collection in Section 5.1, followed by the more extensive collection of articles from the news paper Der Standard in Section 5.2. Some remarks conclude the paper in Section 6. 2 The Self Organizing Map The Self Organizing Map (SOM) as proposed in [2] and described thoroughly in [7, 8, 9] is one of the most distinguished artificial neural network mod els adhering to the unsupervised learning paradigm. The SOM is a general unsupervised tool for the or dering of high dimensional data in such a way that similar items are grouped spatially close to one other. The range of ....

J.A. Kangas, T. Kohonen, and J.T. Laakso- nen, "Variants of self-organizing maps," IEEE

....approach in the sense that the task of segmentation can be achieved without supervision without major shortcomings. Better results should be achieved by considering alternative clustering techniques to determine the classes. This can consider either the LVQ (learning vector quantization) [Kangas et al. 1990)] or the GLVQ (generalized learning vector quantization) Pal et al. 1993) among others. These approaches could lower down the mismatch ratios, improving the classification results. Similar comments must be made about the multiscale approach, which deals with a coarse to fine procedure, which ....

Kangas, J. A.; T. K. Kohonen; and J. T. Laaksonen. Variants of self-organizing maps. IEEE Trans. Neural Networks, 1(1):93-99, 1990.

....and too complex topological structure. Moreover, in Kohonen s model, the size of the network and its neighbourhood topology are chosen at the start and remains xed throughout the learning process. But for the present purpose such a scheme does not work well as for the situations described in [10, 11, 12]. To overcome the limitations of Kohonen s model, it is necessary to implement a dynamic change in the network topology to t it into the present skeletonization task. In the present application we have considered the extension of the Topology Adaptive Self Organizing Neural Network model ....

Kangas JA, Kohonen T and Laaksonen J, Variants of self-organizing maps. IEEE Trans. Neural Networks 1990: 1: 93-99.

....3.2. 2 Neural Network approaches to computing inverse functions Self Organizing Maps Unsupervised algorithms which produce self organizing maps from data have been developed and used by a number of researchers, see, e.g. von der Malsburg 73] Kohonen 82] Ritter, Martinetz Schulten 89] Kangas, Kohonen Laakson 90] Typically these algorithms operate to preserve neighborhoods on a network of nodes which encode the sample 24 data. For example, Kohonen s algorithm is a vector quantization which preserves a neighborhood relation among the nodes, such that areas of high density in the data sample have higher ....

....which preserves a neighborhood relation among the nodes, such that areas of high density in the data sample have higher density of nodes. Convergence results only exist for a small class of topologies of maps; however, the algorithms appear to work well in practice for maps of general topology [Kangas, Kohonen Laakson 90] Applications of SOMs to the robot inverse kinematics problem are given in [Ritter, Martinetz Schulten 89] Martinetz, Ritter Schulten 90] and [Walter Schulten 93] In each of these, either the robot has no redundant dof, or the redundancy is resolved at training time, with only a single ....

J. A. Kangas, Teuvo Kohonen & J. T. Laakson (1990), "Variants of Self--Organizing Maps", IEEE Trans. Neural Networks, 1:1, 93--99.

....[14] has proposed a physiologically plausible method of cooperative and competitive organization for connectionist systems that allows them to self organize around a set of input vectors. Several variants on Kohonen s Self Organizing Topological Feature Maps (Kohonen Maps) have been suggested [12]. These have included networks which allow the learning of output values as well (e.g. for the control of physical systems [22] All of the variants involving output learning, however, have required the use of teachers or critics to provide desired responses or response evaluations (respectively) ....

J. A. Kangas, T. K. Kohonen, and J. T. Laaksonen. Variants of self-organizing maps. IEEE Trans. on Neural Networks, pages 93--99, 1990.

....one center vector is updated. The centers are updated according to the rule j (t 1) j (t) j (X i Gamma j (t) 11) where j is the learning rate and j (t) is the center vector estimate at the moment t. Various decaying rules for the learning rate were tested for the LVQ algorithm [28]. The learning rate which achieves the minimum output variance [29] is j = 1 n j (12) where n j is the number of samples assigned to the cluster j. For the covariance matrix calculation we use the extension of the LVQ algorithm for second order statistics [16, 19] Sigma j (t 1) n j ....

J. A. Kangas, T. K. Kohonen, J. T. Laaksonen, "Variants of self-organizing maps," IEEE Trans. on Neural Networks, vol. 1, no. 1, pp. 93-99, Mar. 1990.

....of the data [15] ffl Instead of the traditional fixed topologies used with SOM (linear and grid) a flexible minimum spanning tree (MST) on the map units is used to define the neighborhood relationships. Such a topology can provide better representation of structured (nonconvex) distributions [12]. In this topology, arcs are assigned between map units and the length of the arcs is defined as the distance between the nodes in the input space. By definition, the sum of the length of all the arcs in the tree is minimal. The sections of the spanning tree between the branching points thus ....

J. A. Kangas, T. K. Kohonen, and J. T. Laaksonen. "Variants of Self-Organizing Maps ". IEEE Transactions on Neural Networks, 1:93--99, 1990.

.... [19] Several design algorithms have been developed for vector quantization [20] 19] 6] 1] 18] 17] The most widely used method for designing vector quantizers is the generalized Lloyd algorithm, known also as LBG scheme [10] 19] An alternative is Kohonen s learning algorithm [3] 4] [2], 5] which can be seen as a generalization of the LBG method [10] The mechanism of the self organizing maps SOM proposed by Kohonen leads to ordering (by linear topology for the cluster units) of multidimensional space which tend to approximate to the density function of the input vectors. In ....

KANGAS J., KOHONEN T. & LAAKSONEN J., Variants of self-organizing maps, IEEE Trans. on Neural Networks 1, 93--99 (1990).

....obtained is parameterized by the topological coordinates of the SOM units. The basic map topology employed in our method is the minimum spanning tree on the map units. The minimum spanning tree provides a flexible topology to represent the input space and its choice is based on empirical results [6] which show that the minimum spanning tree topology is well suited to approximate structured shape distributions. Since a spanning tree is insufficient to describe topological properties like closure of circular regions, we augment the tree structure of the map after convergence with proximity ....

....traditional topologies used with SOM (linear and grid) however, may not suffice to approximate the variation in natural shape distributions. The use of a minimum spanning tree to define the neighborhood relationships in cases where the input vector distribution is structured, has been suggested in [6]. This paper suggests assigning arcs between nodes, where the length of the arcs is defined as the distance between the nodes in the input space. By definition, the sum of the length of all the arcs in the tree is minimal. The neighborhood of a unit in this topology is defined in terms of the arcs ....

J. A. Kangas, T. K. Kohonen, and J. T. Laaksonen. "Variants of Self-Organizing Maps". IEEE Transactions on Neural Networks, 1:93--99, 1990.

....definitions we can write g(z; C; H(Q(z; C 2 ; r k (z) Gamma Q(z; C 2 ; w k (z) 12) where H is the Heaviside unit step function. 13 We will often abbreviate the Q terms in (12) as Q r and Qw . Similar types of terms occur in learning vector quantization (LVQ) and related algorithms [KKL90, KK91, Kos91, GS91]. LVQ is essentially an adaptive nearest neighbor algorithm which sequentially modifies a set of exemplar points. These exemplar points are distinct from the points in the training set. Our algorithm could be extended to give an LVQ like algorithm by incorporating a set of exemplar points and ....

Jari A. Kangas, Teuvo K. Kohonen, and Jorma T. Laaksonen. Variants of selforganizing maps. IEEE Trans. on Neural Nets, 1(1):93--99, March, 1990.

....imply limitations on the resulting mappings. A number of variations have been proposed concerning networks with variable topology or variable number of elements. The approach of Jokusch (1990) leads to networks with rather complicated structure. In the minimumspanning tree network described by Kangas, Kohonen Laaksonen (1990) the preservation of neighborhood relations is done only to a small degree due the sparse connectivity of the network. The neural gas algorithm of Martinetz Schulten (1991) seems to produce compact networks which preserve the neighborhood relations extremely good. It generates, however, in ....

Kangas, J. A., T. Kohonen & T. Laaksonen (1990), Variants of Self-Organizing Maps, IEEE Transactions on Neural Networks, 1, pp. 93--99.

....of the input space. If the learning period is too large computational resources are wasted in needless calculations. The selection of the the learning period s duration of has traditionally been done empirically and Grossberg has been critical of systems that requiring such learning parameters [10]. ffl The model is limited to learning systems with stationary probability density functions. This model is not capable of learning an input domain where the series of sample input vectors have been drawn from a distribution which is varying as a function of time. Although Kohonen has recently ....

....ffl The model is limited to learning systems with stationary probability density functions. This model is not capable of learning an input domain where the series of sample input vectors have been drawn from a distribution which is varying as a function of time. Although Kohonen has recently [10] introduced a dynamically defined neighbourhood in an attempt to overcome this problem it only a partial solution to the problem. These two major limitations of the KFM have been recently removed by the development of the Adaptive Kohonen Feature Map model reported elsewhere [11] ffl Often ....

[Article contains additional citation context not shown here]

Jari Kangas, Teuvo Kohonen, Jorma Laaksonen, Olli Simula, and Olli Vento. Variants of selforganizing maps. In Proceedings of the International Joint Conference on Neural Networks

....of the input space has to match the topology of the output space which is to be represented. In addition, there exist no cost function that yields Kohonon s adaptation rule as its gradient. Some of the above issues have been discussed in the literature by Blackmore et al. 5] Kangas et al. [6], Li et al. 7] Bauer et al. 8] and Fritzke [9] Martinetz et al. 10, 11] proposed the neural gas (NG) network algorithm for vector quantisation, prediction and topology representation a few years ago. The NG network model 1) converges quickly to low distortion errors, 2) reaches a distortion ....

J. A. Kangas, T. K. Kohonen, and J. T. Laaksonen, "Variants of Self-Organizing Maps," IEEE Transactions on Neural Networks, vol. 1, pp. 93--99, Mar. 1990.

....difference between the two implemented algorithms is the definition of this neighborhood . The Extended Self organizing Feature Map algorithm uses fixed neighborhood relations in contrast to the Neural Gas approach which builds these relations dynamical. for an alternative technique see [3]) 1) Neural Gas Algorithm : In the neural gas algorithm the degree of neighborhood is given by the rank of closeness k of the neuron s reference vector to the input vector. For each new target location we assess the sequence ( 0 ; 1 ; N Gamma1 ) of neural units by increasing ....

J.A. Kangas, T. Kohonen, and J.T. Laakson. Variants of self-organizing maps. IEEE Trans. Neural Networks., 1(1):93--99, 1990.

....as well, considering that also folding of the input space into the output space can occur. This case is visualized in Fig. 2, which shows a map of a line onto a square. We should note here, that other map formation algorithms have been proposed, which induce nontrivial output space topologies [17, 18]. As long as the output spaces provide distance measures, the following discussion of the topographic product applies to these cases too. 3 Topographic Product The topographic product is a measure for the preservation of neighborhood relations in maps between spaces of possibly different ....

J.A. Kangas, T.K. Kohonen, J.T. Laaksonen, "Variants of Self-Organizing Maps", IEEE Trans. Neur. Netw. 1, 93-99 (1990).

....in a map of the entire configuration space fiber bundle rather than a single cross section. 3. 3 Proposed Method: Self Organizing Maps of Fibers Unsupervised algorithms which produce self organizing maps from data have been developed and used by a number of researchers, see, e.g. 39] 40] 41] [42]. Typically these algorithms operate to preserve neighborhoods on a network of nodes which encode the sample data. For example, Kohonen s algorithm is a vector quantization which preserves a neighborhood relation among the nodes, such that areas of high density in the data sample have higher ....

....which preserves a neighborhood relation among the nodes, such that areas of high density in the data sample have higher density of nodes. Convergence results only exist for a small class of topologies of maps; however, the algorithms appear to work well in practice for maps of general topology [42]. The general form of a self organizing map consists of a set of simple processing elements (or nodes) which are topologically ordered, perhaps along several dimensions. Each node computes a nonlinear (generally scalar) function of points in the input space. Typically the function computed by ....

J. A. Kangas, Teuvo Kohonen & J. T. Laakson (1990), "Variants of Self--Organizing Maps", IEEE Trans. Neural Networks, 1:1, 93--99.

....be used to add nodes to the structure. The new structure would be re organized, and the process continued. At every epoch, the algorithm would guide the map toward representing the high dimensional properties of the data set accurately. Approaches that employ such heuristics to some extent include [1, 2, 3, 4, 5, 8, 10, 12, 13]. Fritzke s growing cell structure algorithm [1, 2, 3] is particularly interesting because it incorporates methods for both the incremental build up and the periodic correction of the network structure. The basic layout of the map, however, is not a 2 dimensional grid of nodes, but rather a ....

J. Kangas, T. Kohonen, and J. Laaksonen. "Variants of self-organizing maps." IEEE Transactions on Neural Networks, 1:93--99, 1990.

....This shortcoming of the conventional model has been noticed since its introduction, and has been addressed before by several researchers. Kohonen proposed to maintain a minimal spanning tree in the feature map to give a hierarchical representation of the similarity relations of the data [13]. Other approaches proposed include growing cell structures [14] and neural gas networks [15] which can adapt the neighborhood relations dynamically. In this paper, we propose a new approach which makes use of dynamic or adaptive neighborhood relations defined by Gabriel graphs [16] 2 Adaptive ....

J. Kangas, T. Kohonen & J. Laaksonen (1989). Variants of self-organizing maps. IEEE Transactions on Neural Networks, 1:93--99.

....of the existing parallel implementations are discussed. Finally, we draw some conclusions concerning the task of designing parallel computers for SOM. 2. 0 BACKGROUND An overview of the different models of self organizing maps and the application areas where they have been used can be found in [26, 28, 29, 30, 31]. Below we only restate the basic models and refer to the references above for more details. 2.1 Competitive Learning In competitive learning [30, 47] the responses from the adaptive nodes (weight vectors) tend to become localized. After appropriate training the nodes specify clusters or ....

.... by that, Kohonen has developed a class of artificial neural network (ANN) models which develop these, so called, self organizing maps (SOM) also referred to as topological feature maps (TFM) They are all models with competitive learning and use first, second, or higher order topological maps [26]. SOM may be formed with unsupervised learning, i.e. without any teacher saying what is right or wrong. This type of SOM is referred to as self organizing feature maps (SOFM) see Algorithm 3. Algorithm 3 The SOFM algorithm (higher order topology) 1. Find the node (or weight vector) closest to ....

Kangas, J. A., T. K. Kohonen and J. T. Laaksonen. "Variants of self-organizing maps." IEEE Transaction on Neural Networks. Vol. 1(1): pp. 93-99, 1990.

....taken from 3 squares with uniform but different densities. 5. REMARCS ABOUT CONNECTIONS BETWEEN DATA DRIVEN SPACE FILLING CURVES AND KOHONEN S SELF ORGANIZING FEATURE MAPS There exist connections between data driven space filling curves and Kohonen s self organizing maps (SOM) 15] 16] 17] [13]. The mechanism of the self organizing feature maps proposed by Kohonen leads to ordering (by linear topology for the cluster units) of multidimensional space which tend to approximate to the density function of the input vectors. Kohonen claims [17] that in this way one can obtain some ....

KANGAS J., KOHONEN T. & LAAKSONEN J., Variants of self-organizing maps, IEEE Trans. on Neural Networks 1, 93--99 (1990).

No context found.

T. K. Kohonen, J. A. Kangas, and J. T. Laaksonen. Variants of self-organizing maps. Neural Networks, IEEE Transactions on, 1(1):93 -- 99, 1990.

No context found.

Kangas,J.A.,Kohonen,T.K.,Laaksonem,J.T.,1990. Variants of self-organizing maps. IEEE Transactions on Neural Networks 1,93--99.

No context found.

J. A. Kangas, T. Kohonen, and J. Laaksonen, "Variants of self-organizing maps," IEEE Trans. Neural Networks, vol. 1, pp. 93--99, 1990.

No context found.

Kangas, J., Kohonen, T., Laaksonen, J., Simula, O., and Venta, O. (1989). Variants of self-organizing maps. In Proceedings of the International Joint Conference on Neural Networks, pages 517--22, Washington, USA.

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