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Max-margin Clustering: Detecting Margins from Projections of Points on Lines

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Max-margin Clustering: Detecting Margins from Projections of Points on Lines. Raghuraman Gopalan 1 , and Jagan Sankaranarayanan 2 1 Center for Automation Research, University of Maryland, College Park, MD USA 2 NEC Labs, Cupertino, CA USA E-mail: { raghuram,jagan }@umiacs.umd.edu .

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slide1
Max-margin Clustering: Detecting Margins from Projections of Points on Lines

Raghuraman Gopalan1, and Jagan Sankaranarayanan2

1Center for Automation Research, University of Maryland, College Park, MD USA

2NEC Labs, Cupertino, CA USA

E-mail: {raghuram,jagan}@umiacs.umd.edu

slide2
Problem Statement
  • Given an unlabelled set of points forming k clusters, find a grouping with maximum separating margin among the clusters
    • Prior work: (Mostly) Establish feedback between different label proposals, and run a supervised classifier on it
    • Goal: To understand the relation between data points and margin regions by analyzing projections of data on lines
slide3
Two-cluster Problem
  • Assumptions
  • Linearly separable clusters
    • Kernel trick for non-linear case
  • No outliers in data (max margin exist only between clusters)
    • Enforce global cluster balance
  • Proposition 1
  • SI* exists ONLY on line segments in margin region that are perpendicular to the separating hyperplane
    • Such line segments directly provide cluster groupings
slide4
Multi-cluster Problem

SI* doesn’t exist

Location information of projected points (SI) alone is insufficient to detect margins

slide5
The Role of Distance of Projection

Proposition 2

For line intervals in margin region, perpendicular to the separating hyperplane,

Proposition 3

For line intervals inside a cluster of length more than Mm,

Proposition 4

An interval with SI having no projected points with distance of projection less than Dmin*, can lie only outside a cluster; where

γ2

CL2

γ3

CL3

CL1

Defn: Dmin of a line interval is the minimum distance of projection of points in that interval.

No outlier assumption: Max margin between points within a cluster

γ1

slide6
A Pair-wise Similarity Measure for Clustering
  • f(xi,xj)=1, iff xi=xj
  • f(xi,xj)<<1, iff xi and xj are from different clusters, and Intij is perpendicular to their separating hyperplane
slide7
Max-margin Clustering Algorithm
  • Draw lines between all pairs of points
  • Estimate the probability of presence of margins between a pair of points xi and xj by computing f(xi,xj)
  • Perform global clustering using f between all point-pairs
slide9
Summary

ClusteringDetecting margin regions

  • Obtaining statistics of location and distance of projection of points that are specific to line segments in margin regions (Prop. 1 to 4)
  • A pair-wise similarity measure to perform clustering, which avoids some optimization-related challenges prevalent in most existing methods
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