1 / 1

Clustering Detecting margin regions

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 . Multi-cluster Problem.

indiya
Download Presentation

Clustering Detecting margin regions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 Multi-cluster Problem 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 Location information of projected points (SI) alone is insufficient to detect margins Results 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 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 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 Two-cluster Problem Summary • Assumptions • Linearly separable clusters • Kernel trick for non-linear case • No outliers in data (max margin exist only between clusters) • Enforce global cluster balance 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 A Pair-wise Similarity Measure for Clustering • 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 References • 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 F. De la Torre, and T. Kanade, “Discriminative cluster analysis”, ICML, pp. 241-248, 2006. ([8] in table) K. Zhang, I.W. Tsang, and J.T. Kwok, “Maximum margin clustering made practical”, IEEE Trans. Neural Networks, 20(4), pp. 583-596, 2009. ([31] in table)

More Related