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Max-margin Clustering: Detecting Margins from Projections of Points on LinesRaghuraman Gopalan1, and Jagan Sankaranarayanan21Center for Automation Research, University of Maryland, College Park, MD USA2NEC Labs, Cupertino, CA USAE-mail: {raghuram,jagan}@umiacs.umd.edu

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 linesProblem Statement

Two-cluster Problem Proposition 1SI* exists ONLY on line segments in margin region that are perpendicular to the separating hyperplane Such line segments directly provide cluster groupings AssumptionsLinearly separable clusters Kernel trick for non-linear caseNo outliers in data (max margin exist only between clusters) Enforce global cluster balance

Multi-cluster ProblemLocation information of projected points (SI) alone is insufficient to detect marginsSI* doesnt exist

The Role of Distance of ProjectionDefn: 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

Proposition 2For line intervals in margin region, perpendicular to the separating hyperplane,

Proposition 3For line intervals inside a cluster of length more than Mm,

Proposition 4An interval with SI having no projected points with distance of projection less than Dmin*, can lie only outside a cluster; where

CL1CL2CL3123

A Pair-wise Similarity Measure for Clustering f(xi,xj)=1, iff xi=xj f(xi,xj)