Clustering Lecture 9: Other Topics. Jing Gao SUNY Buffalo
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1 Clustering Lecture 9: Other Topics Jing Gao SUNY Buffalo 1
2 Basics Outline Motivation, definition, evaluation Methods Partitional Hierarchical Density-based Miture model Spectral methods Advanced topics Clustering ensemble Clustering in MapReduce Subspace clustering, co-clustering, semi-supervised clustering 2
3 Clustering High-Dimensional Data High-dimensional data everywhere Many applications: tet documents, DNA microarray data Major challenges: Many irrelevant dimensions may mask clusters Distance measure becomes meaningless due to equi-distance Clusters may eist only in some subspaces 3
4 The Curse of Dimensionality (graphs adapted from Parsons et al. KDD Eplorations 2004) Data in only one dimension is relatively packed Adding a dimension stretch the points across that dimension, making them further apart Adding more dimensions will make the points further apart high dimensional data is etremely sparse Distance measure becomes meaningless due to equi-distance 4
5 Why Subspace Clustering? (adapted from Parsons et al. SIGKDD Eplorations 2004) Clusters may eist only in some subspaces Subspace-clustering: find clusters in all the subspaces 5
6 CLIQUE (Clustering In QUEst) Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD 98) Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space Basic idea of CLIQUE It partitions each dimension into the same number of equal length interval It partitions an high dimensional data space into non-overlapping rectangular units A unit is dense if the fraction of total data points contained in the unit eceeds the input model parameter A cluster is a maimal set of connected dense units within a subspace 6
7 CLIQUE: The Major Steps (1) Grid density Partition the data space and find the number of points that lie inside each cell of the partition Dense subspace Identify the subspaces that contain clusters using the Apriori principle Dense subspace in (d+1)-dimension should be dense in d- dimension Start with 1-d units and find the dense units in all the subspaces 7
8 CLIQUE: The Major Steps (2) Identify clusters Determine dense units in all subspaces Determine connected dense units in all subspaces Generate minimal description for the clusters Determine maimal regions that cover a cluster of connected dense units for each cluster Determination of minimal cover for each cluster 8
9 Salary (10,000) age Vacation (week) age Vacation age 9
10 Clustering Definition Revisited n points in R d Group them to k clusters Represent them by a matri A R n d A point corresponds to a row of A Clustering: Partition the rows to k clusters groups R n d n A 10
11 Co-Clustering Co-Clustering Cluster rows and columns of A simultaneously: k = A Co-cluster 11
12 gene Co-Clusters in Gene Epression Data condition Epression level of the gene under the conditions All these genes are activated for some set of conditions Genes Co-cluster Conditions 12
13 K-Means Objective Function Revisited A I A R M Original data points A Data representation A In A, every point in A is replaced by the corresponding cluster center The quality of the clustering is measured by computing distances between the data entries of A and A min j C k ( - m k 2 ) min i j ( A ij - A' ij 2 ) 13
14 Co-Clustering Objective Function A R M C In A every point in A is replaced by the corresponding co-cluster center The quality of the clustering is measured by computing distances between the data in the cells of A and A min i, j ij C ( k ij - m k 2 ) min i j ( A ij - A' ij 2 ) 14
15 Co-Clustering by Bipartite Graph Partitioning Eample Find co-clusters in documents Co-clusters indicate that a set of keywords frequently occur together in a set of documents Bipartite graph formulation Document-word association Bipartite graph partitioning Result partitions are coclusters Docs Words 15
16 Probabilistic Models for Co- Clustering Miture model for clustering first pick one of the components with probability then draw a sample from that component distribution Co-clustering first pick one of the row clusters with probability first pick one of the column clusters with probability then draw a sample from the co-cluster distribution (combination of row and column clusters forms a co-cluster) p r p c 16
17 Input: Semi-supervised Clustering: Problem Definition A set of unlabeled objects, each described by a set of attributes A small amount of domain knowledge Output: A partitioning of the objects into k clusters Objective: Maimum intra-cluster similarity Minimum inter-cluster similarity High consistency between the partitioning and the domain knowledge 17
18 Semi-Supervised Clustering Domain knowledge Partial label information is given Apply some constraints (must-links and cannot-links) Approaches Search-based Semi-Supervised Clustering Alter the clustering algorithm using the constraints Similarity-based Semi-Supervised Clustering Alter the similarity measure based on the constraints Combination of both 18
19 Semi-Supervised K-Means for partially labeled data Seeded K-Means: Labeled data provided by user are used for initialization: initial center for cluster i is the mean of the seed points having label i. Seed points are only used for initialization, and not in subsequent steps. Constrained K-Means: Labeled data provided by user are used to initialize K- Means algorithm. Cluster labels of seed data are kept unchanged in the cluster assignment steps, and only the labels of the nonseed data are re-estimated. 19
20 Seeded K-Means Eample 20
21 Seeded K-Means Eample Initialize Means Using Labeled Data 21
22 Seeded K-Means Eample Assign Points to Clusters 22
23 Seeded K-Means Eample Re-estimate Means 23
24 Seeded K-Means Eample Assign points to clusters and Converge the label is changed 24
25 Constrained K-Means Eample 25
26 Constrained K-Means Eample Initialize Means Using Labeled Data 26
27 Constrained K-Means Eample Assign Points to Clusters 27
28 Constrained K-Means Eample Re-estimate Means and Converge 28
29 COP K-Means COP K-Means [Wagstaff et al.: ICML01] is K-Means with mustlink (must be in same cluster) and cannot-link (cannot be in same cluster) constraints on data points. Initialization Cluster centers are chosen randomly Algorithm During cluster assignment step in COP-K-Means, a point is assigned to its nearest cluster without violating any of its constraints. If no such assignment eists, abort. 29
30 Determine its label COP K-Means Algorithm Must-link Assign to the red class 30
31 Determine its label COP K-Means Algorithm Cannot-link Assign to the red class 31
32 Determine its label COP K-Means Algorithm Must-link Cannot-link The clustering algorithm fails 32
33 Similarity-Based Semi-Supervised Clustering Train an adaptive similarity function to fit the labeled data Use a standard clustering algorithm with the trained similarity function to cluster the unlabeled data Adaptive similarity functions: Altered similarity matri [Kamvar:IJCAI03] Trained Mahalanobis distance [Xing:NIPS02] Altered Euclidian distance [Klein:ICML02] Clustering algorithms: Spectral clustering [Kamvar:IJCAI03] Complete-link agglomerative [Klein:ICML02] K-means [Xing:NIPS02] 33
34 Using Constraints to Alter Similarity Cannot Link Pull points away Must Link Drag points close Must Link Drag points close 34
35 Altered similarity matri Paper: Spectral learning. Kamvar et al. Graph based clustering W: similarity matri D: degree matri (row sum of W) Key idea: alter the similarity matri W based on the domain knowledge 35
36 Semi-supervised spectral clustering 1. Compute the similarity matri W and D 2. For each pair of must-link (i,j), assign W ij = W ji =1 3. For each pair of cannot-link (i,j), assign W ij = W ji = Form the matri D WD 5. Form the matri Y consisting of the first K eigenvectors of D WD 6. Normalize Y so that all the rows have unit lengths 7. Run K-Means on the rows to get the K clusters 36
37 Distance metric learning Paper: Distance metric learning, with application to clustering with side-information. E. Xing, et al. Given two sets of pairs S and D: Compute a distance metric which respects these two sets 37
38 Distance metric learning Define a new distance measure of the form: Linear transformation of the original data 38
39 Original Data A-Diagonal Matri (Rescaling) A-Full Matri (Any transformation) Source: E. Xing, et al. Distance metric learning 39
40 Take-away Message Subspace clustering tries to find clusters in subspaces in high-dimensional data Co-clustering tries to find strong associations among a set of objects with respect to a set of attributes Semi-supervised clustering tries to improve clustering based on eisting domain knowledge (labeled data or pairwise constraints) Many other topics to be eplored for clustering 40
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