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

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]


Problem Statement Points on Lines

  • 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


Two-cluster Problem Points on Lines

  • 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


Multi-cluster Problem Points on Lines

SI* doesn’t exist

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


The Role of Distance of Projection Points on Lines

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


A Pair-wise Similarity Measure for Clustering Points on Lines

  • 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 Algorithm Points on Lines

  • 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


Results Points on Lines

3D

2D


Summary Points on Lines

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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