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<group> |
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<p>The <b>Agglomerative Information Bottleneck (AIB)</b> algorithm |
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greedily compresses discrete data by iteratively merging the two |
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elements which cause the mutual information between the data and the |
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class labels to decreases as little as possible.</p> |
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<p>Here we test AIB on the problem of finding a discriminatively |
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optimal quantization of a mixture of Gaussians. The data in this case |
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is 2 dimensional:</p> |
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<div class="figure"> |
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<image src="%pathto:root;demo/aib_basic_data.jpg"/> |
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<div class="caption"> |
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<span class="content"> |
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Random data generated from a Gaussian mixture with three components |
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(class labels are indicated by color). |
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</span> |
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</div> |
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</div> |
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<p>We quantize this data on a fixed lattice (a 20x20 grid shown in the |
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figures below), and construct histograms for each class.</p> |
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<pre> |
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f1 = quantize(X1,D,K) ; |
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f2 = quantize(X2,D,K) ; |
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f3 = quantize(X3,D,K) ; |
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Pcx(1,:) = vl_binsum(Pcx(1,:), ones(size(f1)), f1) ; |
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Pcx(2,:) = vl_binsum(Pcx(2,:), ones(size(f2)), f2) ; |
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Pcx(3,:) = vl_binsum(Pcx(3,:), ones(size(f3)), f3) ; |
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</pre> |
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<p>Next we apply AIB:</p> |
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<pre> |
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[parents, cost] = vl_aib(Pcx) ; |
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</pre> |
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<p>This provides us with a list of parents of each column |
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in <code>Pcx</code>, forming a tree of merges. We can now "cut" this |
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tree to obtain any number of clusters.</p> |
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<div class="figure"> |
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<image src="%pathto:root;demo/aib_basic_clust_2.jpg"/> |
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<image src="%pathto:root;demo/aib_basic_clust_3.jpg"/> |
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<image src="%pathto:root;demo/aib_basic_clust_4.jpg"/> |
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<div class="caption"> |
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<span class="content"> |
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Three "cuts" of the merge tree, showing 10, 3, and 2 clusters. The |
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gray squares are nodes of the tree which did not have any data points |
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which were quantized to them. |
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</span> |
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</div> |
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</div> |
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<p>Notice that the resulting clusters do not have to be contiguous in |
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the original space.</p> |
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</group> |
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