A particular Gaussian mixture model for clustering and its ...

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H. SahbiTable 1 This table shows the correlations (normalized inner products)between the hyperplanes found, when (i) running Algorithm2 and (ii)solving the QP (6) using the LOQO package, on the 2D toy data ofFig. 4 (σ = 10)LOQO pack vs. Algorithm2 Class 1 Class 2 Class 3 Class 4Cluster 1 0.994 0.043 0.331 0.187Cluster 2 0.036 0.999 0.238 0.133Cluster 3 0.284 0.313 0.985 0.058Cluster 4 0.152 0.147 0.057 0.998Rows characterize the clusters founds after running our algorithm whilecolumns characterize the classes (i.e., ground truth)Table 2 Comparison of the run-time performances on the trainingsamples in Fig. 4 for different values of N (C = 20,σ = 10)#S (N) 40 80 100 500# of parameters in the QP (N × C) 280 480 700 3,500LOQO packAlgorithm224.6 (s)87.9 (s) 317.8 (s)> 1 (H)0.09 (s) 0.32 (s) 0.56 (s) 24.99 (s)encoded using 20 coefficients of projection into a subspacelearned using ISOMAP (Tanenbaum et al. 2000). Notice thatISOMAP embeds a training database into a subspace suchthat non-linearly separable classes become more separable,homogeneous in terms of scale, and easier to cluster.5.1 ScaleIt might be easier for database categorization to predict thevariance of data rather than the number of clusters, mainlyfor large databases living in high dimensional spaces. As thenumber of clusters is initially set to an overestimated value(see Sect. 4.2), our method relies mainly on one parameter,i.e., the scale σ , which makes it possible to find automaticallythe actual number of clusters. In our experiments, we predictσ by sampling manually few images from some categories,estimating the scales of the underlying classes, then settingσ to the expectation of the scale through these classes. Whilethis setting is not automatic, it has at least the advantage ofintroducing a priori knowledge on the variance of the data atFig. 5 These diagrams showthe variation of the scale withrespect to the number ofsamples per class on (left)theOlivetti and (right) Columbiasets. The scale is given as theexpectation of the distancebetween pairs of images takenfrom 3, 5, 7 and 9 classesScale454035302520153 classes5 classes7 classes9 classesScale10.80.60.40.23 classes5 classes7 classes9 classes100 2 4 6 8Number of samples00 10 20Number of samplesFig. 6 (Top) Decrease of thenumber of clusters with respectto the scale parameter. (Bottom)Probability of error with respectto σ . These observations aregiven using (left) the Olivettiand (right) the Columbia setsProbability of errorNumber of Clusters353025201510500 10 20 30 40 50 60Scale0.60.5Olivetti0.40.30.20.1Probability of errorNumber of Clusters25201510500 0.2 0.4 0.6 0.8 1Scale0.6Columbia0.50.40.30.20.100 10 20 30 40 50 60Scale00 0.2 0.4 0.6 0.8 1Scale123
A particular Gaussian mixture model for clustering and its application to image retrievalTable 3 This table shows the distribution of the categories (ground truth) through the clusters on the Olivetti set using our clustering algorithmCategories 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Totalclusters1 9 . . . . . . . . . . . . . . . . . . . 92 . 10 . . 1 3 . . . 1 . . . . . . . . 3 . 183 . . 10 . . . . . . . . . . . . . . 5 . . 154 . . . 10 . . . . 3 . . . . . . . 2 . . . 155 . . . . 9 . . . . . . . . . . . . . . 1 106 . . . . . . 5 . . . . . . . . . . . . . 57 . . . . . . . . 5 . 2 . . . . . . . . . 78 . . . . . 1 . . . 9 . . . . . . . . . 2 129 . . . . . . . . . . 3 . . . . . . . . . 310 1 . . . . . . . . . . 4 5 . 2 . . . . . 1211 . . . . . . . . . . 5 . . . . . . . . . 512 . . . . . . . . . . . 6 3 . . . . . . . 913 . . . . . . . . . . . . 2 10 . 1 . . . . 1314 . . . . . 1 . . . . . . . . 8 . . . . . 915 . . . . . 1 . . . . . . . . . 9 . . . . 1016 . . . . . 4 1 2 2 . . . . . . . 8 2 . . 1917 . . . . . . . . . . . . . . . . . 3 . 5 818 . . . . . . 4 . . . . . . . . . . . 7 . 1119 . . . . . . . . . . . . . . . . . . . 2 2Total 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10We can see that this distribution is concentrated near the diagonal. The scale parameter σ is set to 24 and C = 30. Errors in the cardinality of theclusters are mentioned in boldTable 4 This table shows the distribution of the categories through the clusters on the Columbia set using our clustering algorithmCategories 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Totalclusters1 72 . . . . . . . . . . . . . . 722 . 72 . . . . . . . . . . . . . 723 . . 72 . . . . . . . . . . . . 724 . . . 72 . . . . . . . . . . . 725 . . . . 72 . . . . . . . . . . 726 . . . . . 37 . 34 . . . . . . . 717 . . . . . 35 . 38 . . . . . . . 738 . . . . . . 72 . . . . . . . . 729 . . . . . . . . 72 . . . . . . 7210 . . . . . . . . . 72 . . . . . 7211 . . . . . . . . . . 72 . . . . 7212 . . . . . . . . . . . 72 . . . 7213 . . . . . . . . . . . . 30 . . 3014 . . . . . . . . . . . . 42 72 72 186Total 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72We can see that this distribution is concentrated near the diagonal. The scale σ is set to 0.4andC = 20. Errors in the cardinality of the clusters arementioned in boldthe expense of few interactions and reasonable effort fromthe user.Figure 5 shows that this heuristic provides “good guess” ofthe scale on the Olivetti and Columbia sets as it is close to theoptimal scale shown in Fig. 6. Indeed, the estimated scale inFig. 5 is approximately 24 and 0.4 for, respectively, Olivettiand Columbia sets. For these scales, the number of clustersfound by our algorithm on Olivetti and Columbia (resp. 19123
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