H. Ritter and K. Schulten, "On the stationary state of Kohonen's self-organizing sensory mapping," Biological Cybernetics, vol. 54, pp. 99--106, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of high probability. This means that the correlation between the neurons and their respective weights will be nonlinear. This effect can explicitely be computed for specific cases for the standard SOM: The density of neurons with respect to the underlying data distribution has a specific ratio [38]. Hence the distances of the indices are larger for neurons which lie in regions with a high density of the data compared to regions where only few data, hence few neurons can be observed. What is the effect if recursive data are processed Naturally, the deviation of subtrees which are ....

H. Ritter and K. Schulten. On the stationary state of Kohonen's self-organizing sensory mapping. Biological Cybernetics, 54(1):99-106, 1086.

....maps concerns the so called mag iusually the grid J[ is chosen as a dA dimensional hypercube and then one has i = i, ida) Yet, other ordered arrangements are also admissible. nification. The standard SaM distributes the pointers according to the input distribution p (w) p (v) 3. 7) 2 [11], 12] 23 For the NG one finds, by analytical with the magnification factor asoM = d which only depends considerations, that for small ;k the magnification factor aNG d 2 on the dimensionality of the input space embedded in 1 Dr, i.e. the result is valid for all dimensions [9] Topology ....

H. Ritter, K. Schulten, On the stationary state of Kohonen's self-organizing sensory mapping, Biol. Cyb. 54 (1986) 99-106.

....that the speed of convergence is better than with data obtained by independent random drawings. Another interesting result can be found in [7] and has been completed by many results taking into account the number of neighbours used during learning, the shape of the neighbourhood function, [15, 16]. We know that without weights, the initial distribution cannot be reconstructed exactly. More precisely, 7] shows that the best vector quantization (i.e. which leads to a minimisation of distortion n, without neighbour) corresponds to a discrete density g(x) which converges asymptotically (when ....

Ritter H. and Shulten K., On the Stationary State of Kohonen's Self-Organizing Sensory Mapping, Biol. Cybem., 54, 99-106, 1986.

....(extended to the neighbors) 2(f,q,q2, q, i Uc k x qi 2f(x)dx (5) keVO) where V(i) is the set i 1, i, i 1 . See in Annex A some details about the calculations. See also in the Annex A a very simple method to estimate in particular cases the so called magnification factor ( 4] [10]) We have also to note that the increase of the total processing time of one iteration when using the 2 neighbors SOM algorithm instead of the SCL algorithm is significantly less than 1 . So we propose in the next section to use a mixed algorithm, beginning by a SOM algorithm and ending with a ....

Ritter H. and Shulten K., On the Stationary State of Kohonen's Self-Organizing Sensory Mapping, Biol. Cybern., 54, 99-106, 1986.

....30 35 40 G(i1,5) Fig. 2: Data distribution of the 4 cluster problem. by the neighbourhood function e(d; t) Generally this function reaches its maximum for the winning neuron (i = w ) d = 0) and decreases with the distance from the winner and with the training time t. A typical choice (e.g. in [4, 1]) of the adaptation function and for the decrease of height and width is: e(d; t) ff(t)e Gamma d 2 2oe(t) 2 (3) ff(t) ff init e Gamma t H w(t) oe init e Gamma t W (4) The advantage of an exponential decay is its easy computation and the comprehensive interpretation of the ....

H. Ritter and K. Schulten, "On the stationary state of Kohonen's self-organizing sensory mapping", Biol. Cybern., vol. 54, pp. 99--106, 1986.

....that the speed of convergence is better than with data obtained by independent random drawings. Another interesting result can be found in [7] and has been completed by many results taking into account the number of neighbours used during learning, the shape of the neighbourhood function, [15, 16]. We know that without weights, the initial distribution cannot be reconstructed exactly. More precisely, 7] shows that the best vector quantization (i.e. which leads to a minimisation of distortion x n , without neighbour) corresponds to a discrete density g(x) which converges asymptotically ....

Ritter H. and Shulten K., On the Stationary State of Kohonen's Self-Organizing Sensory Mapping, Biol. Cybern., 54, 99-106, 1986.

....but may also help to understand the formation of sensory maps in the brain. In these maps, the external information is represented in a topology preserving manner, i.e. neighboring units code similar input signals. Properties of the Kohonen learning procedure have been studied in great detail [10, 52]. Most of these studies focussed on the convergence properties of the learning rule, i.e. asymptotic properties of the learning network in a perfectly ordered configuration. In this context, Ritter and Schulten [51, 53] were the first to use the master equation for a description of on line ....

H. Ritter and K. Schulten. On the stationary state of Kohonen's self-organizing sensory mapping. Biological Cybernetics, 54:99--106, 1986.

....finally visualized on the semantic feature map. The final map reflects both the syntactic and the semantic properties of the words. In the self organizing process, the distribution of the weight vectors becomes an approximation of the input vector distribution (Kohonen 1982a, 1989; Ritter 1991; Ritter and Schulten 1986). More weight vectors are allocated to dense areas of the input space, and as a result these areas are magnified (represented to greater detail) on the map. This can be clearly seen in the word maps. For example, the semantic representations for the different animals are very similar, spanning ....

Ritter, H. J., and Schulten, K. J. 1986. On the stationary state of Kohonen's self-organizing sensory mapping. Biological Cybernetics, 54:99--106.

....the neighborhood size. Self organizing feature maps have been applied, for example, into modeling the development of retinotopy, ocular dominance and orientation preference in the visual cortex, and the development of somatotopy in the somatosensory cortex (Obermayer et al. 1990a, 1990b, 1992; Ritter and Schulten 1986). The neocortex contains an extremely dense network of long range connections tangential to the cortical lamination. These connections may mediate cooperation and competition between neurons during self organization of afferent connections (Kohonen 1982, 1989; von der Malsburg and Singer 1988) ....

Ritter, H. J. and Schulten, K. J. (1986). On the stationary state of Kohonen's self-organizing sensory mapping. Biological Cybernetics, 54:99--106.

....i can be formulated as a stochastic on line process, 20] Closely related to stochastic minimization of J and hence appropriate for use as clustering algorithms as well are the various types of self organizing maps. For the original SOM of Kohonen, 21] no energy function exists. Ritter et al. [22], however, have shown that under certain assumptions the distribution of pointers can be described by a magnification factor of = 2=3. On behalf of his Neural Gas algorithm, Martinetz [14] was able to find an energy function closely related to J and to derive the magnification factor of = ....

H. Ritter and K. Schulten, "On the stationary state of kohonen's selforganizing sensory mapping," Biological Cybernetics, vol. 54, pp. 99--106, 1986.

....model [3] is an artificial one, but computationally more effective due to much lower dimensionality of input representation. Both approaches have been used e.g. in modelling the somatotopic map, i.e. projecting the local touch stimuli from the body surface onto the cortex. In the latter approach [4], each touch stimulus was represented by a couple of coordinates of its center of gravity , so the dimension of input space in this case is actually only two. In the former approach [5] a stimulus pattern was modelled by a Gaussian activity profile, centered at some location in input sheet. From ....

H. Ritter and K. Schulten. On the stationary state of kohonen's self-organizing sensory mapping. Biological Cybernetics, 54:99--106, 1986.

....for the cluster units) of multidimensional space which tend to approximate to the density function of the input vectors. In his work [4] Kohonen indicates that linear SOM produce approximations of space filling curves, which mimick probability measures induced by data. In further studies [11], 10] 18] it was etabilished, that in fact this is true only for uniform density of data in one dimension. and the density of the SOM output units in the input space is proportional to the ff power of the input probability density (only in one dimension and if the number of units goes to the ....

RITTER H. & SCHULTEN K., On the stationary state of Kohonen`s selforganizing sensory mapping, Biol. Cybern. 54, 99--106 (1986).

....by data. In fact this is true only for uniform density of data in one dimension. Studies show that the density of the SOM output units in the input space is proportional to the 2=3 power of the input probability density (only in one dimension and if the number of units goes to the infinity) [25], 24] 36] For some application the good output should be proportional exactly to the input probability density [36] The approach proposed in this paper allows to obtain space filling curves which tend to approximate to the density function of the input vectors in multidimensional 0.2 ....

RITTER H. & SCHULTEN K., On the stationary state of Kohonen`s self-organizing sensory mapping, Biol. Cybern. 54, 99--106 (1986).

....neuron i is the best matching unit bmu. The entropy H is then defined as H = Gamma X i2A Pi i Delta log Pi i ; 2. 8) and H becomes maximal if ff = 1 holds [Bra90] However, for the SOM in the case of mapping of a one dimensional input space onto a chain of neurons one gets ff SOFM = 2 3 [RS86] 3 . For the TRN one finds in general ff TRN = k k 2 [MBS93] i.e. ff TRN only depends from the dimensionality of the data set V embedded in the IR k . These results yield that the entropy does not become maximal and, on the other hand, the SOFM minimizes the (somewhat strange) ....

H. Ritter and K. Schulten. On the stationary state of Kohonen's self-organizing sensory mapping. Biol. Cyb., 54:99--106, 1986.

....part of technical systems. This latter class of more abstract map algorithms includes Kohonen s self organizing map algorithm (SOM, for a general discussion see Kohonen, 1995, Ritter et al. 1992) The SOM not only formed the basis of several specific models for the development of biological maps (Ritter, 1986, Obermayer et al. 1990a, Obermayer et al. 1990b, Obermayer et al. 1992, Goodhill, 1993, Wolf et al. 1994, Bauer, 1995) but was also used as a neighborhood preserving vector quantizer in signal processing tasks. The SOM algorithm can be formulated in two different ways, a high dimensional and ....

A 45, 7568-7589. H. Ritter, K. Schulten, 1986. On the stationary state of Kohonen's self-organizing sensory mapping, Biol. Cybern. 54, 99-106.

.... for map development is Kohonen s Self Organizing Map algorithm (SOM) SOM based models have sucessfully accounted for various aspects of visual (Obermayer et al. 1990, 1992, Goodhill, 1993, Wolf et al. 1994, 1996, Bauer et al. 1997) auditory (Martinetz et al. 1988) and somatosensory (Ritter and Schulten, 1986, Andres et al. 1994) map formation. Yet, in simulations of SOM based models for ON center and OFF center cell competition, a break of rotational symmetry has not been observed so far, despite a long lasting search in several groups. This negative outcome could be the consequence of a suboptimal ....

Ritter, H., Schulten, K. (1986). On the stationary state of Kohonen's selforganizing sensory mapping. Biol. Cyb. 54, 99-106.

....Such a selective magnification is not only observed in biological maps, but is also often regarded as a desirable design objective in technical contexts. For at least three reasons, the magnification properties of SOFMs deserve further investigation: 1. An analysis by Ritter and Schulten [10] demonstrated that the SOFM algorithm does not yield a maximum entropy map (i.e. does not transmit the maximum amount of information) 2. As a related argument we observe that it depends on the error criterion one applies which magnification properties are to be regarded as optimal. For example, a ....

....M(w) P (w) i.e. an exponent = 1. On the other hand, in an d dimensional input space, the reconstruction error E p = Z V j w s Gamma v j p dv (3) is minimized (for a large number of neurons) by maps with = d= d p) 12] For SOFMs with d = 1 the analysis by Ritter and Schulten [10] led to = 2=3. 1 So one dimensional SOFMs magnify regions of high stimulation, compromising between an information theoretical optimum ( 1) and a minimal mean square error map ( 1=3) As announced in the introduction we now proceed to modify the original algorithm such that one does not ....

[Article contains additional citation context not shown here]

H. Ritter, K. Schulten, 1986. On the stationary state of Kohonen's self--organizing sensory mapping. Biol. Cybern. 54, 99-106.

.... The relation between the areal magnification factor of the map (given by the receptive field center density P (w) and the stimulus density P (v) can be characterized by an exponent M , P (w) P (v) M : 4) For one dimensional SOFMs an exponent M = 2=3 has been analytically derived [13]. For higher dimensions only a few special cases could be solved [14] As was pointed out in the introductory section, modeling of the perceptual magnet effect requires maps with a negative exponent, such that areas of low intensity of stimulation are are magnified to a high degree (corresponding ....

H. Ritter, K. Schulten, On the stationary state of Kohonen's self-- organizing sensory mapping. Biol. Cybern. 54, 99-106 (1986).

No context found.

Biol Cybern 65:55--63. Ritter H, Schulten K (1986) On the stationary state of Kohonen's self-organizing sensory mapping. Biol Cybern 54:99--106.

....application of the update rule than to leave these states. 4 Energy Functions 4. 1 Energy Functions in One Dimension Does there exist a global energy function such that the convergence to stationary states can be described as a stochastic gradient descent minimizing this potential Following Ritter and Schulten (1986 and 1989) we will introduce average forces acting on the weights and show that, in contrast to the case of a discrete pattern manifold (Ritter 1988) in the general case these forces cannot be derived from a potential function. A global energy function does not exist and the best we can do is ....

Ritter H, Schulten K (1986) On the stationary state of Kohonen's self-organizing sensory mapping. Biol Cybern 54:99--106.

No context found.

H. Ritter and K. Schulten, "On the stationary state of Kohonen's self-organizing sensory mapping," Biological Cybernetics, vol. 54, pp. 99--106, 1986.

No context found.

H. Ritter and K. Schulten. On the stationary state of kohonen's self-organizing sensory mapping. Biological Cybernetics, 54:99--106, 1986.

No context found.

Ritter, H. and K. Schulten (1986). On the stationary state of Kohonen's self-organizing sensory mapping. Biological Cybernetics 54, 99--106.

No context found.

H. Ritter and K. Schulten. On the Stationary State of Kohonen's Self--Organizing Sensory Mapping. Biol. Cybern., 54:99 -- 106, 1986.

No context found.

H. Ritter and K. Schulten. On the stationary state of kohonen's self-organizing sensory mapping. Biological Cybernetics, 54:99--106, 1986.

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