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Efficient Algorithms for Non-Parametric Clustering With Clutter

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Efficient Algorithms for Non-Parametric Clustering With Clutter

Weng-Keen Wong

Andrew Moore

Minefield detection

(Dasgupta and Raftery 1998)

Earthquake faults

(Byers and Raftery 1998)

(Pereira 2002)

(Sloan Digital Sky Survey 2000)

Single Linkage Clustering

Mixture of Gaussians with a Uniform Background Component

Original Dataset

Cuevas-Febrero-Fraiman

(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

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

Find the high

density datapoints

- Cluster the high density points using Single Linkage Clustering
- Stop when link length >

- Originally intended to estimate the number of clusters
- Can also be used to find clusters against a noisy background

A datapoint is a high

density datapoint if:

The number of

datapoints within a

hypersphere of radius

h is > threshold c

- 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)

- Traditional MST algorithms assume you are given all the distances
- Implies O(N2) memory usage
- Want to use a Euclidean Minimum Spanning Tree algorithm

- 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

1. High level overview of GeoMST2

2. Example of a WSPD

3. More detailed description of GeoMST2

4. Our optimizations

1.Create the Well-Separated Pairwise Decomposition

(A1,B1)

(A2,B2)

.

.

.

(Am,Bm)

1.Create the Well-Separated Pairwise Decomposition

Each Pair (Ai,Bi) represents a possible edge in the MST

(A1,B1)

(A2,B2)

.

.

.

(Am,Bm)

1.Create the Well-Separated Pairwise Decomposition

(A1,B1)

(A2,B2)

.

.

.

(Am,Bm)

2.Take the pair (Ai,Bi) 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.

- Let A and B be point sets in d
- Let RA and RB be their respective bounding hyper-rectangles
- Define MargDistance(A,B) to be the minimum distance between RA and RB

The point sets A and B are considered to be

well-separated if:

MargDistance(A,B) max{Diam(RA),Diam(RB)}

Pair #1:

([0],[1])

Pair #2:

([0,1], [2])

Pair #3:

([0,1,2],[3,4])

Pair #4:

([3], [4])

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

A WSPD

(A1,B1)

(A2,B2)

.

.

.

(Am,Bm)

If there are n points, a WSPD can be constructed with O(n) pairs using a fair split tree (Callahan 1995)

1.Create the Well-Separated Pairwise Decomposition

(A1,B1)

(A2,B2)

.

.

.

(Am,Bm)

2.Take the pair (Ai,Bi) 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

Given two sets (Ai,Bi), the Bichromatic

Closest Pair Distance is the closest distance

from a point in Ai to a point in Bi

1.Create the Well-Separated Pairwise Decomposition

(A1,B1)

(A2,B2)

.

.

.

(Am,Bm)

2.Take the pair (Ai,Bi) with the shortest BCP distance

3.If Ai and Bi are not already connected, add the edge to the MST. Repeat Step 2.

Current MST

Current MST

Current MST

Current MST

Current MST

1.Create the Well-Separated Pairwise Decomposition

Modification for CFF:

If BCP distance > , terminate

(A1,B1)

(A2,B2)

.

.

.

(Am,Bm)

2.Take the pair (Ai,Bi) with the shortest BCP distance

3.If Ai and Bi are not already connected, add the edge to the MST. Repeat Step 2.

- 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

Optimizations take place in Step 1

1.Create the Well-Separated Pairwise Decomposition

(A1,B1)

(A2,B2)

.

.

.

(Am,Bm)

2.Take the pair (Ai,Bi) with the shortest BCP distance

3.If Ai and Bi are not already connected, add the edge to the MST. Repeat Step 2.

Ignore all links that are >

- Every pair (Ai, Bi) in the WSPD becomes an edge unless it joins two already connected components
- If MargDistance(Ai,Bi) > , then an edge of length cannot exist between a point in Ai and Bi
- Don’t include such a pair in the WSPD

- Join all elements that are within distance of each other
- If the max distance separating the bounding hyper-rectangles of Ai and Bi is , then join all the points in Ai and Bi if they are not already connected
- Do not add such a pair (Ai,Bi) to the WSPD

- 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

- Ran step two algorithms on subsets of the Sloan Digital Sky Survey
- Compared Kruskal, GeoMST2, and
-clustering

- 7 attributes – 4 colors, 2 sky coordinates, 1 redshift value

- -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