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131 lines
5.7 KiB
HTML
131 lines
5.7 KiB
HTML
<!DOCTYPE html>
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<h1>CURE and TSNE</h1>
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<h2>Clustering Using Representatives (CURE)</h2>
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<p>Clustering using Representatives is a Hierarchical clustering technique in which you can represent a cluster using a <strong>set</strong> of well-scattered representative points.</p>
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<p>This algorithm has a parameter $\alpha$ which defines the factor of the points in which to shrink towards the centroid.</p>
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<p>CURE is known to be robust to outliers and able to identify clusters that have a <strong>non-spherical</strong> shape and size variance.</p>
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<p>The clusters with the closest pair of representatives are the clusters that are merged at each step of CURE's algorithm.</p>
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<p>This algorithm cannot be directly applied to large datasets due to high runtime complexity. Several enhancements were added to address this requirement</p>
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<ul>
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<li>Random sampling: This involves a trade off between accuracy and efficiency. One would hope that the random sample they obtain is representative of the population</li>
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<li>Partitioning: The idea is to partition the sample space into $p$ partitions</li>
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</ul>
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<p>Youtube Video: <a href="https://www.youtube.com/watch?v=JrOJspZ1CUw">https://www.youtube.com/watch?v=JrOJspZ1CUw</a></p>
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<p>Steps</p>
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<ol>
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<li>Pick a random sample of points that fit in main memory</li>
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<li>Cluster sample points hierarchically to create the initial clusters</li>
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<li>Pick representative point<strong>s</strong>
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<ol>
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<li>For each cluster, pick $k$ representative points, as dispersed as possible</li>
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<li>Move each representative points to a fixed fraction $\alpha$ toward the centroid of the cluster</li>
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</ol></li>
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<li>Rescan the whole dataset and visit each point $p$ in the data set</li>
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<li>Place it in the "closest cluster"
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<ol>
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<li>Closest as in shortest distance among all the representative points.</li>
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</ol></li>
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</ol>
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<h2>TSNE</h2>
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<p>TSNE allows us to reduce the dimensionality of a dataset to two which allows us to visualize the data.</p>
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<p>It is able to do this since many real-world datasets have a low intrinsic dimensionality embedded within the high-dimensional space. </p>
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<p>Since the technique needs to conserve the structure of the data, two corresponding mapped points must be close to each other distance wise as well. Let $|x_i - x_j|$ be the Euclidean distance between two data points, and $|y_i - y<em>j|$ he distance between the map points. This conditional similarity between two data points is:
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$$
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p</em>{j|i} = \frac{exp(-|x_i-x_j|^2 / (2\sigma<em>i^2))}{\sum</em>{k \ne i}{exp(-|x_i-x_k|^2/(2\sigma_i^2))}}
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$$
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Where we are considering the <strong>Gaussian distribution</strong> surrounding the distance between $x_j$ from $x_i$ with a given variance $\sigma_i^2$. The variance is different for every point; it is chosen such that points in dense areas are given a smaller variance than points in sparse areas.</p>
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<p>Now the similarity matrix for mapped points are
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$$
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q_{ij} = \frac{f(|x_i - x<em>j|)}{\sum</em>{k \ne i}{f(|x_i - x_k)}}
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$$
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Where $f(z) = \frac{1}{1 + z^2}$</p>
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<p>This has the same idea as the conditional similarity between two data points, except this is based on the <strong>Cauchy distribution</strong>.</p>
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<p>TSNE works at minimizing the Kullback-Leiber divergence between the two distributions $p<em>{ij}$ and $q</em>{ij}$
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$$
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KL(P || Q) = \sum<em>{i,j}{p</em>{i,j} \log{\frac{p<em>{ij}}{q</em>{ij}}}}
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$$
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To minimize this score, gradient descent is typically performed
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$$
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\frac{\partial KL(P||Q)}{\partial y_i} = 4\sum<em>j{(p</em>{ij} - q_{ij})}
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$$</p>
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