k-means Clustering David S. Rosenberg April 24, 2018 New York University
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1 k-means Clustering David S. Rosenberg New York University April 24, 2018 David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
2 Contents 1 k-means Clustering 2 k-means: Failure Cases 3 k-means Formalized David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
3 k-means Clustering David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
4 Example: Old Faithful Geyser Looks like two clusters. How to find these clusters algorithmically? David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
5 k-means: By Example Standardize the data. Choose two cluster centers. From Bishop s Pattern recognition and machine learning, Figure 9.1(a). David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
6 k-means: by example Assign each point to closest center. From Bishop s Pattern recognition and machine learning, Figure 9.1(b). David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
7 k-means: by example Compute new class centers. From Bishop s Pattern recognition and machine learning, Figure 9.1(c). David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
8 k-means: by example Assign points to closest center. From Bishop s Pattern recognition and machine learning, Figure 9.1(d). David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
9 k-means: by example Compute cluster centers. From Bishop s Pattern recognition and machine learning, Figure 9.1(e). David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
10 k-means: by example Iterate until convergence. From Bishop s Pattern recognition and machine learning, Figure 9.1(i). David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
11 k-means Algorithm: Standardizing the data Without standardizing: Blue and black show results of k-means clustering Wait time dominates the distance metric David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
12 David S. Note Rosenberg several (Newpoints York University) have been reassigned DS-GA 1003 from / CSCI-GA black 2567 to blue cluster. April 24, / 19 k-means Algorithm: Standardizing the data With standardizing:
13 k-means: Failure Cases David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
14 k-means: Suboptimal Local Minimum The clustering for k = 3 below is a local minimum, but suboptimal: From Sontag s DS-GA 1003, 2014, Lecture 8. David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
15 k-means Formalized David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
16 k-means: Setting Let X be a space with some distance metric d. Most commonly, X = R d and d(x,x ) = x x. Dataset D = {x 1,...,x n } X. Goal: Partition data D into k disjoint sets C 1,...,C k. The centroid of C i is defined to be µ i = µ(c i ) = arg min µ X x C i d(x,µ) 2. Note: For Euclidean distance on R d, µ(c i ) is the mean of C i. Based on Shalev-Shwartz and Ben-David s book Understanding Machine Learning, Ch 22. David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
17 k-means: Objective function The k-means objective is J k-means (C 1,...,C k ) = k i=1 = min x C i d(x,µ(c i )) 2 µ 1,...,µ k X i=1 k x C i d(x,µ i ) 2 In vector quantization, we represent each x C i by the centroid µ i. We can think of this as lossy data compression, the k-means objective can be viewed as the reconstruction error. How many bits does it take to represent each point with vector quantization? If k = 2 d, then d bits. (Fewer on average if the clusters have unequal sizes.) David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
18 k-means: Algorithm input: D = {x 1,...,x d } X initialize: Randomly choose initial centroids µ 1,...,µ k D. repeat until convergence (i.e. until the centroids or clusters repeat): i, let C i = { x D : i = arg min j d(x,µ j ) }. (break ties in some arbitrary manner) i, let µ i = argmin µ X x C i d(x,µ) 2. (For Euclidean distance, µ i = 1 C i x C i x) David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
19 k-means++ In k-means, objective never increases, but no guarantee to find minimizer. General recommendation is to re-run with several random starting initial centroids. k-means++ is a way to randomly initialize the centroids with some guarantees: Randomly choose first centroid from the data points D. For each of the remaining k 1 centroids: Compute distance from each x i to the closest already chosen centroid. Randomly choose next centroid with probability proportional to the computed distance squared. If we let Jk-means be the minimizer of the k-means objective, then using k-means++ for initialization guarantees that E[J k-means (C 1,...,C k )] 8(logk + 2)J k-means. David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, / 19
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