Clustering Introduction Machine Learning. Andrew Rosenberg

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1 Clustering Introduction Machine Learning Andrew Rosenberg 1 / 55

2 Clustering Clustering Clustering is an unsupervised Machine Learning application The task is to group similar entities into groups. 2 / 55

3 We do this all the time 3 / 55

4 We do this all the time 4 / 55

5 We do this all the time 5 / 55

6 We can do this in many dimensions 6 / 55

7 We can do this to many degrees 7 / 55

8 We can do this in many dimensions 8 / 55

9 We do this all the time 9 / 55

10 How do we set this up computationally? In Machine Learning, we optimize objective functions to find the best solution. Maximum Likelihood (for Frequentists) Maximum A Posteriori (for Bayesians) Empirical Risk Minimization Loss function Minimization What makes a good cluster? How do we define loss or likelihood in a clustering solution? 10 / 55

11 Cluster Evaluation Intrinsic Evaluation Evaluate the compactness of the clusters Extrinsic Evaluation Compare the results to some gold standard labeled data. (Not covered today) 11 / 55

12 Intrinsic Evaluation Intercluster Variability (IV) How different are the data points within the same cluster Extracluster Variability (EV) How different are the data points that are in distinct clusters Minimize IV while maximizing EV. Minimize IV EV IV = C d(x, c) x C d(x, c) = x c 12 / 55

13 Degenerate Clustering Solutions One Cluster 13 / 55

14 Degenerate Clustering Solutions N Clusters 14 / 55

15 Clustering Approaches Hierarchical Clustering Partitional Clustering 15 / 55

16 Hierarchical Clustering Recursive Partitioning 16 / 55

17 Hierarchical Clustering Recursive Partitioning 17 / 55

18 Hierarchical Clustering Recursive Partitioning 18 / 55

19 Hierarchical Clustering Recursive Partitioning 19 / 55

20 Hierarchical Clustering Recursive Partitioning 20 / 55

21 Hierarchical Clustering Agglomerative Clustering 21 / 55

22 Hierarchical Clustering Agglomerative Clustering 22 / 55

23 Hierarchical Clustering Agglomerative Clustering 23 / 55

24 Hierarchical Clustering Agglomerative Clustering 24 / 55

25 Hierarchical Clustering Agglomerative Clustering 25 / 55

26 Hierarchical Clustering Agglomerative Clustering 26 / 55

27 Hierarchical Clustering Agglomerative Clustering 27 / 55

28 Hierarchical Clustering Agglomerative Clustering 28 / 55

29 Hierarchical Clustering Agglomerative Clustering 29 / 55

30 Hierarchical Clustering Agglomerative Clustering 30 / 55

31 Hierarchical Clustering Agglomerative Clustering 31 / 55

32 K-Means Clustering K-Means clustering is a Partitional Clustering Algorithm. Identify different partitions of the space for a fixed number of clusters Input: a value for K the number of clusters. Output: the K centers of clusters centroids 32 / 55

33 K-Means Clustering 33 / 55

34 K-Means Clustering Algorithm: Given an integer K specifying the number of clusters. Initialize K cluster centroids Select K points from the data set at random Select K points from the space at random For each point in the data set, assign it to the cluster whose center it is closest to. argmin Ci d(x, C i ) Update the centroid based on the points assigned to the cluster. c i = 1 C i x C i x If any data point has changed clusters, repeat. 34 / 55

35 Why does K-Means Work? When an assignment is changed, the sum of squared distances of the data point to its assigned cluster is reduced. IV is reduced. When a cluster centroid is moved the sum of squared distances of the data points within that cluster is reduced IV is reduced. At convergence we have found a local minimum of IV 35 / 55

36 K-Means Clustering 36 / 55

37 K-Means Clustering 37 / 55

38 K-Means Clustering 38 / 55

39 K-Means Clustering 39 / 55

40 K-Means Clustering 40 / 55

41 K-Means Clustering 41 / 55

42 K-Means Clustering 42 / 55

43 K-Means Clustering 43 / 55

44 K-Means Clustering 44 / 55

45 K-Means Clustering 45 / 55

46 K-Means Clustering 46 / 55

47 Soft K-Means In K-means, we forced every data point to be the member of exactly one cluster. We can relax this constraint. p(x, C i ) = d(x, c i) j d(x, c j) p(x, C i ) = Based on minimizing entropy of cluster assignment. exp{ d(x, c i)} j exp{ d(x, c j)} We still define a cluster by a centroid, but we calculate the centroid as a weighted center of all the data points. x c i = x p(x, C i) x p(x, C i) Convergence is based on a stopping threshold rather than changing assignments. 47 / 55

48 Potential Problems with K-Means Optimal? K-means approaches a local minimum, but this is not guaranteed to be globally optimal. Could you design an approach which is globally optimal? Consistent? Different starting clusters can lead to different cluster solutions 48 / 55

49 Potential Problems with K-Means Optimal? K-means approaches a local minimum, but this is not guaranteed to be globally optimal. Could you design an approach which is globally optimal? Sure, in NP. Consistent? Different starting clusters can lead to different cluster solutions 49 / 55

50 Suboptimality in K-Means 50 / 55

51 Inconsistency in K-Means 51 / 55

52 Inconsistency in K-Means 52 / 55

53 Inconsistency in K-Means 53 / 55

54 Inconsistency in K-Means 54 / 55

55 More Clustering Gaussian Mixture Models 55 / 55

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