1. 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

2. 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

3. 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

4. Multi-cluster Problem
SI* doesn’t exist
Location information of projected points (SI) alone is insufficient to detect margins

5. 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 γ1 γ2 γ3
CL1
CL2
CL3
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

6. 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

7. 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

8. Results 3D 2D

9. Clustering Detecting margin regions
Summary
Clustering Detecting 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