Hans-Ulrich Bauer, R. Der, and M. Herrman. Controlling the magnification factor of self-organizing feature maps. Neural Computation, 8:757--771, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of a vector quantizer, proposed in [18] is called the mechanism of conscience yielding approximately the same winning probability for each neuron. Hence, the approach yields density matching between input and output space which corresponds to a magnification factor of 1. BAUER et al. in [19] suggested a more general scheme for the SOM to achieve an arbitrary magnification. They introduced a local learning parameter es for each neuron r with (es cr 7 (Wr) m in (3.5) where m is an additional control parameter. Equation (3.5) now reads as AWr = eshx (r,s,v) v Wr) 3.10) Note that ....

....that the data are highly correlated, and therefore a drastic dimensionality reduction may be possible. However, a faithful topographic mapping is nec essary to preserve the information contained in the hyperspectral image. In addition to the conscience algorithm explicit magnification control [19] according to (3.10) and the growing san (GsaN) procedure, as extensions of the standard saN, are suitable tools [64] The GSON produced a lattice of dimensions 8 x 6 x 6 for the LCVF image, which represents a radical dimension reduction, and it is in agreement with the Grassberger Procaccia ....

H.-U. Bauer, R. Der, M. Herrmann, Controlling the magnification factor of self- organizing feature maps, Neural Computation 8 (4) (1996) 757-771.

....y 8 9; 8 being the intrinsic data dimension. However, a maximum of mutual information is obtained for a magnification 1 [9] 10] Yet, the standard procedure may be extended in such a way that one is able to control the magnification. In analogy to a control scheme for the SOM [11] it was proven that the local learning rates chosen to be i A yield for the new adaptation rule ji0 3k 4 2 42 k z the magnification 1]n B1 kF C d [12] Thereby, is a new additional control parameter to achieve an arbitrary magnification. 3.2 The Supervised ....

H.-U. Bauer, R. Der, M. Herrmann, Controlling the magnification factor of self--organizing feature maps, Neural Computation 8 (4) (1996) 757--771.

....the reconstructed density is shown to be related to f(x) through an exponent as in (6) We insist on the fact that definitions of g(x) are different in [7] and [8] and that further investigation would be necessary to compare these two results. Finally, it must be mentioned that several authors [17, 6] proposed modifications of the algorithm to compensate the exponent in (6) and to obtain discrete densities of centroids closer fromf(x) To our opinion, the question is to know if the exponent in (6) must be considered as an undesired effect which must be compensated, or if the problem of ....

Bauer H.-U., Der R. and Heronann M., Controlling the Magnification Factor of Self-Organizing Feature Maps, Neural Computation, 8,757-771, 1996.

....dimension. with the magnification factor oz Dff 2 Dff However, a maximum of mutual information is obtained for a magnification oz = 1 [9] 10] Yet, the standard procedure may be extended in such a way that one is able to control the magnification. In analogy to a control scheme for the SOM [11] it was proven that the local learning rates chosen to be i P (wi) yield for the new adaptation rule Xw: 8. A (i, v, w) v w) the magnification oz; oz (rn 1) 12] Thereby, rn is a new additional control parameter to achieve an arbitrary magnification. 3.2 The Supervised Neural Gas ....

H.-U. Bauer, R. Der, M. Herrmann, Controlling the magnification factor of self-organizing feature maps, Neural Computation 8 (4) (1996) 757-771.

....the reconstructed density is shown to be related to f(x) through an exponent as in (6) We insist on the fact that definitions of g(x) are different in [7] and [8] and that further investigation would be necessary to compare these two results. Finally, it must be mentioned that several authors [17, 6] proposed modifications of the algorithm to compensate the exponent in (6) and to obtain discrete densities of centroids closer from f(x) To our opinion, the question is to know if the exponent in (6) must be considered as an undesired effect which must be compensated, or if the European ....

Bauer H.-U., Der R. and Herrmann M., Controlling the Magnification Factor of Self-Organizing Feature Maps, Neural Computation, 8, 757-771, 1996.

....is that due to neighborhood cooperation they usually converge much faster than their stochastic counterparts without neighborhood cooperation. A further advantage of stochastic quantizers with neighborhood cooperation is the possibility to actually control the magnification factor as suggested in [23]. 3.2 An ID estimation procedure In order to use tpca for ID estimation we must eventually decide how many dominant eigenvalues exist in each local region, i.e. what size an eigenvalue 4 This follows from the more general result in [17] stating that the reconstruction error E = R M jx ....

H. U. Bauer, "Controlling the magnification factor of self-organizing feature maps," Neural Computation, vol. 8, pp. 757--771, 1996.

....models PFMs and SFMs depend on the cluster density in the CBR L;k (S) that is controlled with the contraction parameter k 2 (0; 1 2 ] Smaller k s yield more dense clusters. Furthermore, quantization of the CBR L;k (S) is controlled by the magnification factor (Ritter, Schulten, 1986; Bauer, Der, Herrmann, 1996) of the used vector quantization scheme. The magnification factor relates 18 the frequency of codebook vectors in a quantized region with the frequency of points from CBR L;k (S) in that region. One can find a formal relationship among the CBRcontraction factor k, magnification factor of the ....

Bauer, H.U., Der, R., & Herrmann, M. (1996). Controlling the magnification factor of self-organizing feature maps. Neural Computation, 8, 757--771.

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H.-U. Bauer, R. Der, and M. Herrmann. Controlling the magnification factor of selforganizing feature maps. Neural Computation, 8(4):757--771, 1996.

....map a data point to the nearest codebook vector resp. receptive field vector An important measure of performance for vector quantization is the reconstruction error. As far as neural maps are concerned, the reconstruction error has been evaluated and compared to other vector quantization schemes [4, 5, 6]. The outcome of such comparisons is often a close call, depends on the particular tasks at hand, and shall not be the main interest in the present paper. Here, we focus on the other prominent feature of neural maps, their topography. This property distinguishes neural maps and neural map ....

H.-U. Bauer, R. Der, M. Herrmann (1995). Controlling the Magnification Factor of Self-Organizing Feature Maps. Neural Computation 8, 757-771.

....of an error measure, principles to govern parameter settings are often more implicit and mainly left to the programmer s experience. Moreover, most general purpose algorithms allow for various definitions of optimality of the resulting network function in accordance to the learning task [1]. Hence, we found it useful to decide on both the algorithm and the task when approaching the problem of parameter learning. More specifically, the self organized feature map algorithm was applied to the task to construct principal manifolds of data distributions. For a related problem in which ....

....the adaptation of particularly local parameters (cf. e.g. 7] Finally, our adaptive algorithm provides a stable version of earlier attempts (cf. 12] pp. 244 250) to construct a principal manifold in a generally high dimensional data set. As a first step in this direction of parameter learning [1] the learning rate of a feature map algorithm was adjusted in order to match various optimality criteria. Here, we concentrate on the more interesting case of locally modifying the neuron s neighborhood ranges. This algorithm, too, can fit to various criteria, such as uniqueness of representation ....

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H.-U. Bauer, R. Der, M. Herrmann (1995) Controlling the magnification factor of selforganizing feature maps. To appear in Neural Computation.

....SOFM algorithm which allows to actually control the magnifcation exponents. A description of this modification in the second section is followed by a report on numerical and analytical results in the third section and a brief discussion. A longer version of this work will be published elsewhere [2]. 2 SOFMs with Local Learning Rates Kohonen s self organizing feature maps project stimuli (or data points) in some input space V onto neurons located at the vertices r of an output space grid A. The map is given in terms of pointers w r 2 V which are associated to each neuron r. A particular ....

....step. 3 Results for One Dimensional Maps In a first test case, we apply our algorithm to maps of the unit interval onto chains of neurons. Here, an analysis of the learning rule (4) and (5) can be performed (analogous to the original derivation in Ref. 10] details will be published elsewhere [2]) and leads to a modified exponent 0 , M(w) P (w) 0 = P (w) 2 3 (1 m) 8) Chosing m = 0, the magnification 0 = 2=3 of the regular SOFM results, chosing m = 1=2 yields an optimally information preserving map with 0 = 1. m = Gamma1=2 yields 0 = 1=3, which in turn ....

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H.-U. Bauer, R. Der, M. Herrmann, 1995. Controlling the magnification factor of self-organizing feature maps. Submitted to Neural Computation.

....X . A nearest neighbor rule assigns x 2 X to r 3 2 R if the distance kx 0 w r k is minimal for r = r 3 . The formation of the maps is achieved by updating the reference vectors, 1w r;i = ffl r 3h rr 3 (x i 0w r;i ) i = 1; nX (1) where ffl r 3 is a unit dependend learning rate [10] and h rr 3 is a Gaussian function of the distance kr 0 r 3 k that imposes the topological structure in R to the reference vectors w r 2 X . If the components of x are uncorrelated there exists also a stationary configuration w r with uncorrelated components w r;i . Whether or not such a ....

....if both distributions factorize [15] This condition is satisfied if the maps separates the sources. Hence, the scale of the output of the map is determined by the input distribution. At least in a linear mixture equidistancy within each source can be preserved if = 0. According to Ref. [10] such a compensation is achieved if ffl r 1=P (x) which results in the case of discrete input in a mere monitoring of the winning frequency, i.e. ffl r 1=P (r) For biasses b 1=2 the performance remained virtually unchanged compared to the unbiased case. This does not happen when constant ....

H.-U. Bauer, R. Der, and M. Herrmann, "Controlling the Magnification Factor of Self-Organizing Feature Maps," Neural Computation, vol. 8, pp. 757-771, 1996.

No context found.

H.-U. Bauer, R. Der, M. Herrmann (1995) Controlling the magnification factor of selforganizing feature maps. Submitted to Neur. Comp.

....rule: map a data point to the nearest codebook vector resp. receptive field vector An important measure of performance for vector quantization is the distortion error. As far as neural maps are concerned, the distortion error has been evaluated and compared to other vector quantization schemes [4], 5] 6] The outcome of such comparisons is often a close call, depends on the particular tasks at hand, and shall not be of further interest in the present paper. Here, we focus on the other prominent feature of neural maps, their topography. The neighborhood relations of data points are ....

H.-U. Bauer, R. Der, M. Herrmann, Controlling the Magnification Factor of Self-Organizing Feature Maps. Neural Computation 8, 757-771, 1995.

No context found.

Hans-Ulrich Bauer, R. Der, and M. Herrman. Controlling the magnification factor of self-organizing feature maps. Neural Computation, 8:757--771, 1996.

No context found.

H.-U. Bauer, R. Der, and M. Herrmann. Controlling the magnification factor of self-organizing feature maps. Neural Computation, 8(4):757--771, 1996.

No context found.

H.-U. Bauer, R. Der, and M. Herrmann. Controlling the magnification factor of self-organizing feature maps. Neural Computation, 8(4):757--771, 1996.

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