H.-U. Bauer, T. Villmann, Growing a Hypercubical Output Space in a Self-Organizing Feature Map, ICSI Tech Rep . TR-95-030 (1995).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the input patterns are again organized in one single at SOM . The shortcoming of having to de ne the size of the SOM in advance has been addressed by a number of di erent models. Consider, for example the Incremental Grid Growing [4] Growing Grid [5] Growing SOM [6] and the Hypercubical SOM [24]. The rst, i.e. Incremental Grid Growing, allows the addition of new units at the boundary of the map. Furthermore, connections between units of the map may be established and removed according to some threshold settings based on the similarity of their respective model vectors. This may result ....

H.-U. Bauer and T. Villmann, \Growing a hypercubical output space in a self-organizing feature map," IEEE Transactions on Neural Networks, vol. 8, no. 2, pp. 226 - 233, 1997.

....and the structure of the lattice precisely coincide with the dimensionality and the structure of the training data. Hence the above measures can be used in order to check whether this property holds. If not, a different lattice structure or the use of adaptive methods like the growing SOM [5] are necessary. The neural gas algorithm or topology representing networks, respectively [33, 34] do not use a prior lattice structure but they explicitely define the lattice structure a posteriori such that similar point are mapped to similar 28 neurons: Those neurons become neighbored which ....

H.-U. Bauer and T. Villmann. Growing a Hypercubical Output Space in a Self--Organizing Feature Map. IEEE Transactions on Neural Networks, 8(2):218--226, 1997.

....processing neurons and connections are added depending on input data. Another solution is the Growing Hierarchical SaM (GHSaM) 3] where clusters of processing neurons are added in a hierarchical manner, allowing quicker classification of input patterns. Also, the Growing SaM (GSaM) algorithm [4] grows within a two dimensional neuron space using the regular SaM algorithm with neurons added using an additional criterion. These methods remove the need for prior specification of the network architecture but assume that the pattern set is finite and that exemplar versions of the patterns are ....

H.-U. Bauer, Th. Villmann "Growing a Hypercubical Output Space in a SelfOrganising Feature Map" ICSI, 1995

....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 error lower than that resulting ....

H.-U. Bauer and T. Villmann, "Growing a Hypercubical Output Space in a Self-Organizing Feature Map," IEEE Transactions on Neural Networks, vol. 8, pp. 218--226, Mar. 1997.

....: Non stationary data distributions can be found in many technical, biological or economical processes. Self organizing neural networks have rarely been considered for tracking those distributions since many of the models, e.g. the selforganizing map [6] neural gas [8] or the hypercubical map [1], use decaying adaptation parameters 1 . Once the adaptation strength has decayed, the network is frozen and thus unable to react to subsequent changes in the signal distribution. 2 : even for incremental networks Models with small constant parameters such as the incremental networks ....

H.-U. Bauer and Th. Villmann. Growing a hypercubical output space in a selforganizing feature map. TR-95-030, International Computer Science Institute, Berkeley, 1995.

....grid symmetrically in all directions therefore ending up always with the same height width ratio of the grid which they used initially. Thus no tuning of this parameter to the data was done as in our approach. Very recently Bauer and Villmann proposed a scheme for growing hypercubical featuremaps [8]. They, too, insert single rows and columns based on statistical measures of the data. In contrast to the growing grid method the insertion of a new row or column is always done in the center of the existing structure no matter which units have received most input signals in the past. Perhaps ....

H.-U. Bauer and Th. Villmann. Growing a hypercubical output space in a self-organizing feature map. Tr-95-030, International Computer Science Institute, Berkeley, 1995.

....in Ray H. White (1992) Competitive Hebbian Learning: Algorithms and Demonstrations, Neural Networks 5, pp. 261 275 Recently Bauer and Villmann have proposed an incremental self organizing network (GSOM) generating also a hyper rectangular structure by inserting complete rows or columns [2]. Instead of using accumulated error info to determine where to insert new units, they always insert near the center of the current topology. The network parameters are time varying (e.g. the neighborhood adaptation is cooled down according to saw tooth shaped function) In contrast to our GG ....

H.-U. Bauer and Th. Villmann. Growing a hypercubical output space in a self-organizing feature map. Tr-95-030, International Computer Science Institute, Berkeley, 1995.

No context found.

H.-U. Bauer and T. Villmann. Growing a Hypercubical Output Space in a Self--Organizing Feature Map. IEEE Transactions on Neural Networks, 8(2):218--226, 1997.

No context found.

H.-U. Bauer and T. Villmann. Growing a hypercubical output space in a selforganizing feature map. IEEE Transactions on Neural Network 8(2):218-226, 1997.

....topology of the neural architecture mirrors the intrinsic topology of the data. Hence various heuristics exist to measure the degree of topology preservation, to adapt the topology to the data, to define the lattice a posteriori, or to evolve structures which are appropriate for real world data [2, 7, 20, 27, 37]. In all tasks the intrinsic dimensionality of data plays a crucial role since it determines an important aspect of the optimum neural network: the topological structure, i.e. the lattice for SOM. Moreover, superfluous data dimensions slow down the training for LVQ as well. They may even cause a ....

.... with neural methods [15, 25] A Grassberger Procaccia analysis estimates the dimensionality of attractors in a dynamic system [12] SOMs which adapt the dimensionality of the lattice during training like the growing SOM (GSOM) automatically determine the approximate dimensionality of the data [2]. Naturally, all adaptation schemes which determine weighting factors or relevance terms for the input dimensions constitute an alternative method for determining the dimensionality: The dimensions which are ranked as least important, i.e. they possess the smallest relevance terms, can be dropped. ....

[Article contains additional citation context not shown here]

H.-U. Bauer and T. Villmann. Growing a Hypercubical Output Space in a Self-- Organizing Feature Map. IEEE Transactions on Neural Networks, 8(2):218--226, 1997.

....neuron is avai]ab]e we use IIw w II or w w to compute the backprojection of the output space direction ei into the input IIw w11 space. 11 After each growth step, a new learning phase has to take place, in order to readjust the map. For a detailed study of the algorithm we refer to [23]. The Growing TRN adapts the number of neurons in TRN whereby the structure between them is re arranged according to the TRN definition [24] Hence, it is capable of representing the data space in a topologically faithful manner with increasing accuracy and it also realizes a structure ....

H.-U. Bauer, T. Villmann, Growing a Hypercubical Output Space in a Self-Organizing Feature Map, IEEE Transactions on Neural Networks 8 (2) (1997) 218-226.

....of the neural architecture mirrors the intrinsic topology of the data. Hence various approaches exist in order to measure the degree of topology preservation, to adapt the topology to the data, to define the lattice a posteriori, or to evolve structures which are appropriate for real world data [2,6,14,19,25]. In all tasks the intrinsic dimensionality of the data plays a crucial role since it determines large parts of the optimum neural network, i.e. the lattice for SOM. Moreover, superfluous data dimensions slow down the training for LVQ as well. They may even cause a decrease in accuracy since they ....

H.-U. Bauer and T. Villmann. Growing a Hypercubical Output Space in a Self-- Organizing Feature Map. IEEE Transactions on Neural Networks, 8(2):218--226, 1997.

....as a function of DA . The zero crossing at DA = 2 indicates that the DA = 2 maps are topographically optimal. 4.2 Noisy non linear manifold The second example is still illustrative and artificial, but more challenging. It has first been described in another investigation of topographic maps [37]. Two parameters x and y, with uniform distribution in 0 x 1, 0 y 0:3, are used to construct 10 dimensional stimuli v with v 1 = x 0:5; v 2 = y 0:3; v 3 = 4xy(1 Gamma x) 13) So the data are sampled from a curved two dimensional manifold embedded in a three dimensional space. The seven ....

....dimensionality. So we trained DA = 2 maps with 64 Theta 4, 28 Theta 9 and 16 Theta 16 output spaces. All measures picked out the 28 Theta 9 maps as the optimal ones. This coincides quite well with the aspect ratio of the underlying parameter space. From an investigation of this example in [37] we also know that the 28 Theta 9 maps visually appear rather undistorted, while the other maps exhibit some amount of topographic distortion. The topography values of the other DA = 2 maps were still substantially better than those for maps with other output dimensions. All measures also allowed ....

[Article contains additional citation context not shown here]

H.-U. Bauer, Th. Villmann (1997). Growing a Hypercubical Output Space in a Self-Organizing Feature Map. IEEE Transactions on Neural Networks 8:2, 226-233.

....optimal. Phi(k 0) These effects do not prohibit the unequivocal identification of the correct dimension DA = 2. B. Noisy non linear manifold The second example is still illustrative and artificial, but more challenging. It has first been described in another investigation of topographic maps [32]. Two parameters x and y, with uniform distribution in 0 x 1, 0 y 0:3, are used to construct 10 dimensional stimuli v with v 1 = x 0:5; v 2 = y 0:3; v 3 = 4xy(1 Gamma x) 14) So the data are sampled from a curved two dimensional manifold embedded in a three dimensional space. The seven ....

....levels above = 0:1. 28 Theta 9 and 16 Theta 16 output spaces. All measures (including Goodhill s minimal wiring) picked out the 28 Theta 9 maps as the optimal ones. This coincides quite well with the aspect ratio of the underlying parameter space. From an investigation of this example in [32] we also know that the 28 Theta 9 maps visually appear rather undistorted, while the other maps exhibit some amount of topographic distortion. The topography values of the other DA = 2 maps were still substantially better than those for maps with other output dimensions. In the presence of low ....

H.-U. Bauer, Th. Villmann, Growing a Hypercubical Output Space in a Self-Organizing Feature Map. Submitted to IEEE Transactions on Neural Networks, 1995.

....the algorithm. 3 Results of Simulations We now present results of simulations on a more challenging data set, which is based on speech data, and which we used in previous publications already [1, 9] The results of more simulations, with more real world data sets, are given in another publication [2]. The present data set has been acquired in the III. Physikalisches Institut, Universitat Gottingen. It contains the ten (German) digits, spoken ten times each by one speaker. The preprocessing of the data resulted in 2013 feature vectors, in a 19 d input space. For this data set the growth ....

Bauer, H.-U.; Villmann, Th., Growing a Hypercubical Output Space in a Self-Organizing Feature Map, ICSI Tech Rep. TR-95-030 (1995).

No context found.

H.-U. Bauer, T. Villmann, Growing a Hypercubical Output Space in a Self-Organizing Feature Map, ICSI Tech Rep . TR-95-030 (1995).

No context found.

Hans-Ulrich Bauer and Thomas Willmann. Growing a hypercubical output space in a self-organizing feature map. IEEE Transaction on Neural Networks, 8(2):218--226, 1997.

No context found.

H.-U. Bauer and T. Villmann, "Growing a hypercubical output space in a self-organizing feature map," IEEE Transactions on Neural Networks, vol. 8, no. 2, pp. 218--226, 1997.

No context found.

H.-U. Bauer and T. Villmann. Growing a Hypercubical Output Space in a Self-Organizing Feature Map. IEEE Transactions on Neural Networks, 8(2):218--226, 1997.

No context found.

H.-U. Bauer and T. Villmann. Growing a hypercubical output space in a self-organizing feature map. IEEE Transactions on Neural Networks, 8(2):218--226, 1997.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC