HES-SO Master of Science in Engineering. Clustering. Prof. Laura Elena Raileanu HES-SO Yverdon-les-Bains (HEIG-VD)
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1 HES-SO Master of Science in Engineering Clustering Prof. Laura Elena Raileanu HES-SO Yverdon-les-Bains (HEIG-VD)
2 Plan Motivation Hierarchical Clustering K-Means Clustering 2
3 Problem Setup Arrange items into clusters High similarity between objects in the same cluster Low similarity between objects in different clusters Cluster labeling is a separate problem 3
4 Applications Exploratory analysis of large collections of objects Image segmentation Recommender systems Cluster hypothesis in information retrieval Computational biology and bioinformatics Pre-processing for many other algorithms 4
5 Two approaches Hierarchical K-Means 5
6 Distance Metrics Non-negativity: Identity: Symmetry: d(x, y) 0 Triangle Inequality: d(x, y) = 0 () x=y d(x, y) = d(y, x) d(x, y) apple d(x, z) + d(z, y) 6
7 Distance: Norms Given: Euclidean distance (L2-norm) Manhattan distance (L1-norm) d(x, y) = d(x, y) = Lr-norm x=[x 1,x 2,...x n ] y=[y 1,y 2,...y n ] d(x, y) = v ux t n (x i y i ) 2 i=0 nx x i y i i=0 " X n # 1/r x i y i r i=0 7
8 Distance: Cosine Given: Idea: measure distance between the vectors Thus: x=[x 1,x 2,...x n ] y=[y 1,y 2,...y n ] cos = x y x y sim(x, y) = pp n i=0 x2 i P n i=0 x iy i pp n i=0 y2 i d(x, y) = 1 sim(x, y) 8
9 Hierarchical Agglomerative Clustering Start with each document in its own cluster Until there is only one cluster: Find the two clusters ci and cj, that are most similar Replace ci and cj with a single cluster ci cj The history of merges forms the hierarchy 9
10 Hierarchical Clustering A B C D E F G H 10
11 Cluster Merging Which two clusters do we merge? What s the similarity between two clusters? Single Link: similarity of two most similar members Complete Link: similarity of two least similar members Group Average: average similarity between members 11
12 Link Functions Single link: Uses maximum similarity of pairs: Can result in straggly (long and thin) clusters due to chaining effect Complete link: Use minimum similarity of pairs: sim(c i,c j ) = sim(c i,c j )= Makes more tight spherical clusters max sim(x, y) x2c i,y2c j min sim(x, y) x2c i,y2c j 12
13 K-Means Algorithm Let d be the distance between documents Define the centroid of a cluster to be: µ(c) = 1 c X x x2c Select k random instances {s1, s2, sk} as seeds. Until clusters converge: Assign each instance xi to the cluster cj such that d(xi, sj) is minimal Update the seeds to the centroid of each cluster For each cluster cj, sj =μ (cj) 13
14 K-Means Clustering Example Pick seeds Reassign clusters Compute centroids Reassign clusters Compute centroids Reassign clusters Converged 14
15 Basic MapReduce Implementation er to. To 5) p- ey ed rewe p- be he st es as er 15 1: class Mapper 2: method Configure() 3: c LoadClusters() 4: method Map(id i, pointp) 5: n NearestClusterID(clusters c, pointp) 6: p ExtendPoint(point p) 7: Emit(clusterid n, pointp) 1: class Reducer 2: method Reduce(clusterid n, points[p 1, p 2,...]) 3: s InitPointSum() 4: for all point p 2 points do 5: s s + p 6: m ComputeCentroid(point s) 7: Emit(clusterid n, centroid m) Algorithm 5: K-means clustering algorithm.
16 we apbe he est zes as er ers orly rk the erion ity. to to Afthe K m, N M the we at ing is HES-SO MSE MapReduce Implementation w/ IMC 16 2: method Reduce(clusterid n, points[p 1, p 2,...]) 3: s InitPointSum() 4: for all point p 2 points do 5: s s + p 6: m ComputeCentroid(point s) 7: Emit(clusterid n, centroid m) Algorithm 5: K-means clustering algorithm. 1: class Mapper 2: method Configure() 3: c LoadClusters() 4: H InitAssociativeArray() 5: method Map(id i, pointp) 6: n NearestClusterID(clusters c, pointp) 7: p ExtendPoint(point p) 8: H{n} H{n} + p 9: method Close() 10: for all clusterid n 2 H do 11: Emit(clusterid n, pointh{n}) 1: class Reducer 2: method Reduce(clusterid n, points[p 1, p 2,...]) 3: s InitPointSum() 4: for all point p 2 points do 5: s s + p 6: m ComputeCentroid(point s) 7: Emit(clusterid n, centroid m) Algorithm Big Data Analytics 6: K-means Clustering Année clustering 2014/2015 algorithm with IMC (in-mapper combining) design pattern.
17 Implementation Notes Standard setup of iterative MapReduce algorithms Driver program sets up MapReduce job Waits for completion Checks for convergence Repeats if necessary Must be able keep cluster centroids in memory With large k, large feature spaces, potentially an issue Memory requirements of centroids grow over time Variant: k-medoids 17
18 Reference Jimmy Lin and Chris Dyer, «Data-Intensive Text Processing with MapReduce», Morgan & Claypool Publishers,
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