1 Efficient Algorithms for Non-parametric Clustering With Clutter Weng-Keen Wong Andrew Moore (In partial fulfillment of the speaking requirement)

2 Problems From the Physical Sciences Minefield detection (Dasgupta and Raftery 1998) Earthquake faults (Byers and Raftery 1998)

3 Problems From the Physical Sciences (Pereira 2002)(Sloan Digital Sky Survey 2000)

4 A Simplified Example

5 Clustering with Single Linkage Clustering ClustersSingle Linkage Clustering MST

6 Clustering with Mixture Models Resulting ClustersMixture of Gaussians with a Uniform Background Component

7 Clustering with CFF Cuevas-Febrero-FraimanOriginal Dataset

8 Related Work (Dasgupta and Raftery 98) Mixture model approach – mixture of Gaussians for features, Poisson process for clutter (Byers and Raftery 98) K-nearest neighbour distances for all points modeled as a mixture of two gamma distributions, one for clutter and one for the features Classify each data point based on which component it was most likely generated from

9 Outline 1. Introduction: Clustering and Clutter 2. The Cuevas-Febreiro-Fraiman Algorithm 3. Optimizing Step One of CFF 4. Optimizing Step Two of CFF 5. Results

10 The CFF Algorithm Step One Find the high density datapoints

11 The CFF Algorithm Step Two Cluster the high density points using Single Linkage Clustering Stop when link length > 

12 The CFF Algorithm Originally intended to estimate the number of clusters Can also be used to find clusters against a noisy background

13 Step One: Density Estimators Finding high density points requires a density estimator Want to make as few assumptions about underlying density as possible Use a non-parametric density estimator

14 A Simple Non-Parametric Density Estimator A datapoint is a high density datapoint if: The number of datapoints within a hypersphere of radius h is > threshold c

15 Speeding up the Non-Parametric Density Estimator Addressed in a separate paper (Gray and Moore 2001) Two basic ideas: 1. Use a dual tree algorithm (Gray and Moore 2000) 2. Cut search off early without computing exact densities (Moore 2000)

16 Step Two: Euclidean Minimum Spanning Trees (EMSTs) Traditional MST algorithms assume you are given all the distances Implies O(N 2 ) memory usage Want to use a Euclidean Minimum Spanning Tree algorithm

17 Optimizing Clustering Step Exploit recent results in computational geometry for efficient EMSTs Involves modification to GeoMST2 algorithm by (Narasimhan et al 2000) GeoMST2 is based on Well-Separated Pairwise Decompositions (WSPDs) (Callahan 1995) Our optimizations gain an order of magnitude speedup, especially in higher dimensions

18 Outline for Optimizing Step Two 1. High level overview of GeoMST2 2. Properties of a WSPD 3. How to create a WSPD 4. More detailed description of GeoMST2 5. Our optimizations

19 Intuition behind GeoMST2

20 Intuition behind GeoMST2

21 High Level Overview of GeoMST2 (A 1,B 1 ) (A 2,B 2 ). (A m,B m ) Well-Separated Pairwise Decomposition

22 High Level Overview of GeoMST2 (A 1,B 1 ) (A 2,B 2 ). (A m,B m ) Well-Separated Pairwise Decomposition Each Pair (A i,B i ) represents a possible edge in the MST

23 High Level Overview of GeoMST2 (A 1,B 1 ) (A 2,B 2 ). (A m,B m ) 1.Create the Well- Separated Pairwise Decomposition 2.Take the pair (A i,B i ) that corresponds to the shortest edge 3.If the vertices of that edge are not in the same connected component, add the edge to the MST. Repeat Step 2.

24 A Well-Separated Pair (Callahan 1995) Let A and B be point sets in  d Let R A and R B be their respective bounding hyper-rectangles Define MargDistance(A,B) to be the minimum distance between R A and R B

25 A Well-Separated Pair (Cont) The point sets A and B are considered to be well-separated if: MargDistance(A,B)  max{Diam(R A ),Diam(R B )}

26 Interaction Product The interaction product between two point sets A and B is defined as: A  B = {{p,p’} | p  A, p’  B, p  p’}

27 Interaction Product The interaction product between two point sets A and B is defined as: A  B = {{p,p’} | p  A, p’  B, p  p’} This is the set of all distinct pairs with one element in the pair from A and the other element from B

28 Interaction Product Definition The interaction product between two point sets A and B is defined as: A  B = {{p,p’} | p  A, p’  B, p  p’} For Example: A = {1,2,3}B = {4,5} A  B = {{1,4}, {1,5}, {2,4}, {2,5}, {3,4}, {3,5}}

29 Interaction Product A  B = {{0,1}, {0,2}, {0,3},{0,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}} Now let A and B be the same point set ie. A = {0,1,2,3,4}B = {0,1,2,3,4}

30 Interaction Product A  B = {{0,1}, {0,2}, {0,3}, {0,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}} Now let A and B be the same point set ie. A = {0,1,2,3,4}B = {0,1,2,3,4} Think of this as all possible edges in a complete, undirected graph with {0,1,2,3,4} as the vertices

31 A Well-Separated Pairwise Decomposition Pair #1: ([0],[1]) Pair #2: ([0,1], [2]) Pair #3: ([0,1,2],[3,4]) Pair #4: ([3], [4]) Claim: The set of pairs {([0],[1]), ([0,1], [2]), ([0,1,2],[3,4]), ([3], [4])} form a Well-Separated Decomposition.

32 Interaction Product Properties If P is a point set in  d then a WSPD of P is a set of pairs (A i,B i ),…,(A k,B k ) with the following properties: 1. A i  P and B i  P for all i = 1,…,k 2. A i  B i =  for all i = 1, …, k A = {0,1,2,3,4}B = {0,1,2,3,4} {([0],[1]), ([0,1], [2]), ([0,1,2],[3,4]), ([3], [4])} clearly satisfies Properties 1 and 2

33 Interaction Product Property 3 3. (A i  B i )  (A j  B j ) =  for all i,j such that i  j From {([0],[1]), ([0,1], [2]), ([0,1,2],[3,4]), ([3], [4])} we get the following interaction products: A 1  B 1 = {{0,1}} A 2  B 2 = {{0,2},{1,2}} A 3  B 3 = {{0,3},{1,3},{2,3},{0,4},{1,4},{2,4}} A 4  B 4 = {{3,4}} These Interaction Products are all disjoint

34 Interaction Product Property 4 4. P  P = {{0,1}, {0,2}, {0,3}, {0,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}} A 1  B 1 = {{0,1}} A 2  B 2 = {{0,2},{1,2}} A 3  B 3 = {{0,3},{1,3},{2,3},{0,4},{1,4},{2,4}} A 4  B 4 = {{3,4}} The Union of the above Interaction Products gives back P  P

35 Interaction Product Property 5 5. A i and B i are well-separated for all i=1,…,k

36 Two Points to Note about WSPDs Two distinct points are considered to be well-separated For any data set of size n, there is a trivial WSPD of size (n choose 2)

37 A Well-Separated Pairwise Decomposition (Continued) If there are n points in P, a WSPD of P can be constructed in O(nlogn) time with O(n) elements using a fair split tree (Callahan 1995)

38 A Fair Split Tree

39 Creating a WSPD Are the nodes outlined in yellow well-separated? No.

40 Creating a WSPD Recurse on children of node with widest dimension

41 Creating a WSPD Recurse on children of node with widest dimension

42 Creating a WSPD Recurse on children of node with widest dimension

43 Creating a WSPD And so on…

44 Base Case Eventually you will find a well-separated pair of nodes. Add this pair to the WSPD.

45 Another Example of the Base Case

46 Creating a WSPD FindWSPD(W,NodeA,NodeB) if( IsWellSeparated(NodeA,NodeB)) AddPair(W,NodeA,NodeB) else if( MaxHrectDimLength(NodeA) < MaxHrectDimLength(NodeB) ) Swap(NodeA,NodeB) FindWSPD(W,NodeA->Left,NodeB) FindWSPD(W,NodeA->Right,NodeB)

47 High Level Overview of GeoMST2 (A 1,B 1 ) (A 2,B 2 ). (A m,B m ) 1.Create the Well- Separated Pairwise Decomposition 2.Take the pair (A i,B i ) that corresponds to the shortest edge 3.If the vertices of that edge are not in the same connected component, add the edge to the MST. Repeat Step 2

48 Bichromatic Closest Pair Distance Given two sets (A i,B i ), the Bichromatic Closest Pair Distance is the closest distance from a point in A i to a point in B i

49 High Level Overview of GeoMST2 (A 1,B 1 ) (A 2,B 2 ). (A m,B m ) 1.Create the Well- Separated Pairwise Decomposition 2.Take the pair (A i,B i ) with the shortest BCP distance 3.If A i and B i are not already connected, add the edge to the MST. Repeat Step 2.

50 GeoMST2 Example Start Current MST

51 GeoMST2 Example Iteration 1 Current MST

52 GeoMST2 Example Iteration 2 Current MST

53 GeoMST2 Example Iteration 3 Current MST

54 GeoMST2 Example Iteration 4 Current MST

55 High Level Overview of GeoMST2 (A 1,B 1 ) (A 2,B 2 ). (A m,B m ) 1.Create the Well- Separated Pairwise Decomposition 2.Take the pair (A i,B i ) with the shortest BCP distance 3.If A i and B i are not already connected, add the edge to the MST. Repeat Step 2. Modification for CFF: If BCP distance > , terminate

56 Optimizations We don’t need the EMST We just need to cluster all points that are within  distance or less from each other Allows two optimizations to GeoMST2 code

57 High Level Overview of GeoMST2 (A 1,B 1 ) (A 2,B 2 ). (A m,B m ) 1.Create the Well- Separated Pairwise Decomposition 2.Take the pair (A i,B i ) with the shortest BCP distance 3.If A i and B i are not already connected, add the edge to the MST. Repeat Step 2. Optimizations take place in Step 1

58 Recall: How to Create the WSPD

59 Optimization 1 Illustration

60 Optimization 1 Ignore all links that are >  Every pair (A i, B i ) in the WSPD becomes an edge unless it joins two already connected components If MargDistance(A i,B i ) > , then an edge of length  cannot exist between a point in A i and B i Don’t include such a pair in the WSPD

61 Optimization 2 Illustration

62 Optimization 2 Join all elements that are within  distance of each other If the max distance separating the bounding hyper-rectangles of A i and B i is  , then join all the points in A i and B i if they are not already connected Do not add such a pair (A i,B i ) to the WSPD

63 Implications of the optimizations Reduce the amount of time spent in creating the WSPD Reduce the number of WSPDs, thereby speeding up the GeoMST2 algorithm by reducing the size of the priority queue

64 Results Ran step two algorithms on subsets of the Sloan Digital Sky Survey 7 attributes – 4 colors, 2 sky coordinates, 1 redshift value Compared Kruskal, GeoMST2, and  -clustering

65 Results (GeoMST2 vs  -Clustering vs Kruskal in 4D)

66 Results (GeoMST2 vs  -Clustering in 3D)

67 Results (GeoMST2 vs  -Clustering in 4D)

68 Results (Change in Time as  changes for 4D data)

69 Results (Increasing Dimensions vs Time

70 Future Work More accurate, faster non-parametric density estimator Use ball trees instead of fair split tree Optimize algorithm if we keep h constant but vary c and 

71 Conclusions  -clustering outperforms GeoMST2 by nearly an order of magnitude in higher dimensions Combining the optimizations in both steps will yield an efficient algorithm for clustering against clutter on massive data sets