Clustering. Barna Saha
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1 Clustering Barna Saha
2 The Problem of Clustering Given a set of points, with a no;on of distance between points, group the points into some number of clusters, so that members of a cluster are close to each other, while members of different clusters are far.
3 Example: Clusters x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x 3
4 Clustering in Low Dimensional Euclidean Space is Easy
5 Modern Clustering Problem May involve Euclidean spaces of very high dimension. Non Euclidean space: Jaccard distance, Cosine Distance, Hamming Distance, Edit Distance etc. Example: Cluster documents by topics based on occurrences of unusual words Cluster moviegoers by the type or types of movies they like Cluster genes by their sequence similarity
6 A Popular Clustering Algorithm: K Means k=number of clusters Given k and a set of data points in the Euclidean space select k centers so as the the sum of squared distance between each point and its nearest center is minimized.p Solving this problem exactly in NP Hard 25 years ago Llyod proposed a simple local seacrh based algorithm that is s;ll very widely used---has polynomial ;me smoothed complexity. However Llyod s algorithm may get stuck at a local op;ma K Means
7 Llyod s Local Search Algorithm
8 Illustra;on (taken from Wiki) Convergence to local op;ma
9 Convergence to Local Op;ma
10 Selec;ng centers by distance works some ;me
11 Selec;ng centers by distance works some ;me
12 Selec;ng centers by distance works some ;me
13 Selec;ng centers by distance works some ;me
14 Sensi;ve to Outlier
15 K-Means Just the ini;aliza;on in Llyod s algorithm changes everything else remains the same.
16 Objec;ve based clustering K-means: K-median K-center
17 K-median In Llyod s algorithm use the next cluster center as the median of the elements in the cluster.
18 Another simple local search algorithm Start with arbitrary k centers: C Assign points to the nearest center and compute the objec;ve value If swapping a vertex x outside of C with a vertex v in C, decreases the objec;ve value, swap The algorithm converges and gives a 5-approxima;on. Be_er approxima;on bound known.
19 K-center Minimizes the maximum distance A simple greedy algorithm gives a 2- approxima;on Pick any vertex v arbitrarily and declare it as the first center For i=2 to k Select the vertex in V that is farthest from the already chosen centers and make it the new i-th center.
20 Clustering with unknown k Say we want to cluster n objects of some kind (documents, images, text strings) But we don t have a meaningful way to project into Euclidean space. Using past data train up some classifier f(x,y)=same/different. Then run f on all pairs and try to find most consistent clustering. 20
21 The problem Harry Bovik Harry B. H. Bovik Tom X. 21
22 The problem Harry Bovik Harry B. : Same -: Different H. Bovik Tom X. Train up f(x)= same/different Run f on all pairs 22
23 The problem Harry Bovik Harry B. : Same -: Different H. Bovik Tom X. Totally consistent: 1. edges inside clusters 2. edges outside clusters 23
24 The problem Harry Bovik Harry B. : Same -: Different H. Bovik Tom X. Train up f(x)= same/different Run f on all pairs 24
25 The problem Harry Bovik Disagreement H. Bovik Harry B. Tom X. : Same -: Different Train up f(x)= same/different Run f on all pairs Find most consistent clustering 25
26 The problem Harry Bovik Disagreement H. Bovik Harry B. Tom X. : Same -: Different Train up f(x)= same/different Run f on all pairs Find most consistent clustering 26
27 The problem Harry Bovik Disagreement H. Bovik Tom X. Problem: Given a complete graph on n ver;ces. Each edge labeled or -. Goal is to find par;;on of ver;ces as consistent as possible with edge labels. Max #(agreements) or Min #( disagreements) Harry B. : Same -: Different There is no k : # of clusters could be anything 27
28 The Problem Noise Removal: There is some true clustering. However some edges incorrect. S;ll want to do well. Agnos;c Learning: No inherent clustering. Try to find the best representa;on using hypothesis Eg: Research communi;es via collabora;on graph 28
29 Nice features of formula;on There s no k. (OPT can have anywhere from 1 to n clusters) If a perfect solu;on exists, then it s easy to find: C(v) = N (v). [Why?] Easy to get agreement on ½ of edges. [Why?] 29
30 Minimizing Disagreements Goal: Get a constant factor approx. 30
31 Minimizing Disagreement Pick a random permuta;on of ver;ces Select v from the random order and create a cluster with all its posi;ve neighbors Remove that cluster with all associated edges Repeat Gives a 3-approxima;on
32 δ-clean Clusters Given a clustering, vertex δ-good if few disagreements C N - (v) Within C < δ C N (v) Outside C < δ C v is δ-good : Similar -: Dissimilar 32
33 Algorithm 1. Pick vertex v. Let C(v) = N (v) 2. Modify C(v) (a) Remove 3δ-bad ver;ces from C(v). (b) Add 7δ good ver;ces into C(v). 3. Delete C(v). Repeat un;l done, or above always makes empty clusters. 4. Output nodes ler as singletons. 33
34 Lower bounding idea: bad triangles Consider We know any clustering has to disagree with at least one of these edges. 34
35 Lower bounding idea: bad triangles If several such disjoint, then mistake on each one Edge disjoint Bad Triangles (1,2,3), (3,4,5) 4 3 D opt >= #{Edge disjoint bad triangles} 35
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