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 [email protected] .

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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

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,[email protected]


Max margin clustering detecting margins from projections of points on lines

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


Max margin clustering detecting margins from projections of points on lines

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


Max margin clustering detecting margins from projections of points on lines

Multi-cluster Problem

SI* doesn’t exist

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


Max margin clustering detecting margins from projections of points on lines

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


Max margin clustering detecting margins from projections of points on lines

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


Max margin clustering detecting margins from projections of points on lines

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


Max margin clustering detecting margins from projections of points on lines

Results

3D

2D


Max margin clustering detecting margins from projections of points on lines

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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