### Table 4. Performance on 100 geometric problems where clusters are inapplicable.

2006

"... In PAGE 8: ...28 variables. As Table4 indicates, all three of our applicable methods solved these problems in 20000 steps, and did so faster, despite the preliminary time required to detect crucial subproblems. Purely random problems should not display struc- ture.... ..."

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### Table 4. Performance on 100 geometric problems where clusters are inapplicable.

2006

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### Table 1. Characterization of selected clustering algorithms with respect to geometrical and geometry-related properties [2, 18, 19].

2003

"... In PAGE 4: ... On the other hand, k-Means is not able to identify clusters that deviate much from a spherical shape. Table1 lists typical geometrical properties of prominent clustering algorithms. 2.... ..."

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### Table 2 shows the average test error per cluster, the relative cluster size N=m, and the geometrical property Nd of cluster size N times the average distance d within the cluster. Given the per-cluster statistic, we would clearly prefer the model 4-9-2.

"... In PAGE 6: ...ents (see Figure 2). Dendrograms 4-7-2 and 4-8-2 display compact clusters. The cluster structure begins to decay in dendrogram 4-9-2, and has completely disappeared in 4-10-2 and the following dendrogram. Table2 : Per-cluster statistic... In PAGE 6: ...election method can, e. g. be applied to Boltzmann machines. In our opinion, if a series of weight vectors does not contain a single cluster of similar solutions, the experiment setup (data set, learning algorithm, or network model) is likely to be inadequate. We conjecture that a purely geometrical approach might be su cient to select adequate models (see Table2 ). De ning a suitable measure for the presence and compactness of clusters might eventually lead to an algorithm that autofocuses to an appropriate network structure.... ..."

### Table 4. The geometrical mean percentage of clusterers selected by selective voting and selective weighted-voting under different ensemble sizes. ensemble size percentage of selecting

### Table 1. Length of Monte Carlo runs for the four-, five- and six-dimensional Ising models, in millions of sampled configurations (MS). Before taking each sample, one Metropolis sweep and a number GC (also shown) of geometrical cluster steps were executed. In a few cases the values of MS and GC shown here represent a weighted average over runs with a different number of cluster steps.

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### Table 3: Clusters possibly related to cluster 4 (Level: 1e-0). Clusters are sorted by quality (i.e. the minus log of the geometric average of connections). Note that all clusters belong to the super family of Immunoglobulins

"... In PAGE 6: ... The alignments can be found in the web site. Table3 is an illustrative example. It shows the clusters which are possibly related to cluster 4 (Im- munoglobulin V region), ordered by quality value.... ..."