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Arbitrary distance function estimation using vector quantization

Arbitrary distance function estimation using vector quantization,10.1109/ICNN.1995.487272,B. John Oommen,I. Kuban Altinel,N. Aras

Arbitrary distance function estimation using vector quantization   (Citations: 1)
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In this paper we shall utilize the concepts of vector quantization (VQ) for the computation of arbitrary distance functions-a problem which has been receiving much attention in the operations research and location analysis community. The input to our problem is the set of coordinates of a large number of nodes whose inter-node arbitrary “distances” have to be estimated. Unlike traditional operations research methods, which use parametric functional estimators, we have utilized VQ principles to first adaptively polarize the nodes into sub-regions according to Kohonen's self-organizing map. Subsequently, the parameters characterizing the sub-regions are learnt by using a variety of methods
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    • ...In two earlier works [5], [47], we demonstrated that the principles of VQ could be utilized naturally and powerfully to solve the arbitrary distance estimation problem...
    • ...Indeed, the solution proposed in [5] and [47] was a sequence of pattern recognition and polarizing modules governed by the laws of VQ. The salient contribution of this present paper is that we have shown that by merging the learning principles of two families of adaptive algorithms we can achieve an enhanced superior learning algorithm...
    • ...However, in the method proposed in [5], [47], the region of interest is subdivided into a set of subregions adaptively using a VQ method, and in our current work this has been done by only restricting ourselves to “integer” points on the grid...
    • ...Although, perceptronbased nonparametric estimators perform better compared to parametric distance functions, (i.e., they yield smaller errors), the results can be improved further if the cities are clustered adaptively using a VQ [5], [47] DVQ method prior to any estimation attempt...
    • ...Note that this can be seen to be the discretized version of the traditional SOM strategy [21], [24], [25], [30], [37] except that we have (as in [5], [47]) consistently restricted the radius of the “bubble of interest” used by Kohonen to be unity...
    • ...This is as recommended in the literature [24], [25] and has been justified in the continuous domain [5], [47]...
    • ... of interest, This restriction has also been recommended in the literature [24], [25], and typically, this window, , is a hypersphere centered at the bisector between the codebook vectors and Also, as recommended in the literature, the polarizing of both and (when both of them correctly classify ) is made to be of much smaller magnitude than in the scenario when either of them misclassifies it. These steps are formally given in [5] and [47] ...
    • ...In all the experiments reported in earlier publications involving Turkey [4], [5], [47], the original map of Turkey was enclosed within a bounding rectangle defined between the latitudes and longitudes 36 N, 26 E, 42 N, and 45 E, respectively...
    • ...Also, in the interest of comparing the current discretized work with its continuous counterpart [5], [47], the results which we report and the initial partitions are exactly the same as those for which we had reported earlier results in [5], [47]...
    • ...Also, in the interest of comparing the current discretized work with its continuous counterpart [5], [47], the results which we report and the initial partitions are exactly the same as those for which we had reported earlier results in [5], [47]...
    • ...This should be compared with the results for the continuous VQ scheme [5], [47] where the most conservative case (obtained by averaging in the -space) yielded a testing error of 7.69, and in the case when the functions are characterized by the test error was 7.12...
    • ...Observe too that like the continuous scheme [5], [47] the most time consuming phase of the learning is the optimization stage...
    • ...Generally speaking, the accuracy is comparable to the other reported schemes (other than the continuous VQ scheme [5], [47]) for small values of This accuracy increases remarkably with the magnification as increases from 2 to 8 and then tends to stabilize thereafter...
    • ...Note that in the random case cited for the continuous VQ algorithm [5], [47], the corresponding errors were 1.787 and 7.189, respectively...
    • ...Unlike for the continuous VQ algorithm [5], [47] we have not been able to determine any initial 2-partitions with six codebook vectors in each which can yield superior classification and testing...
    • ...or hybrid strategy and are often superior even to the case when continuous VQ was used for the polarizing [5], [47]...
    • ...Finally, arguing as in [5], [47], we believe that the VQ and its discretized counterpart are superior to the perceptron-based methods because unlike the latter, the distance function itself is defined on a welldefined Euclidean space...

    B. John Oommenet al. Discrete vector quantization for arbitrary distance function estimatio...

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