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Core-Sets and Geometric Optimization problems.

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## Core-Sets and Geometric Optimization problems.

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**Core-Sets and Geometric Optimization problems.**Piyush Kumar Department of Computer Science Florida State University http://piyush.compgeom.com Email: piyush@acm.org Joint work with Alper Yildirim**Talk Outline**• Introduction to Core-Sets for Geometric Optimization Problems. • Problems and Applications • Minimum Enclosing Balls (Next talk) • Axis Aligned Minimum Volume Ellipsoids • Motivation • Optimization Formulation/IVA • Algorithm • Computational Experiments • Future Directions.**r***Geometric Clustering • In order to cluster, we need • Points ( Polyhedra / Balls / Ellipsoids ? ) • A distance measure • Assume Euclidian unless otherwise stated. • A method for evaluating our clustering. • (We look at k-centers, 1-Ecenter, 1-center, • kernel 1-center, k-line-centers)**Fitting/Covering Problems**• For some shapes, the problem is convex and hence tractable. (MEB / MVE / AAMVE). • Minimize the maximum distance • O(n) in 2D but becomes harder in higher dimensions. • Least squares / SVM Regression / … • d ≠ O(1)?**Fitting multiple subspaces/shapes?**• Non-Convex (Min the max distance) / Non-Linear / NP-Hard to approx. • Has many applications • Minimize sum of distances (orthogonal) : SVD • Other assumptions : GPCA/PPCA/PCA…**Core-Sets**Core Sets are a small subset of the input points whose fitting/covering is “almost” same as the fitting/covering of the entire input. [AHV06 Survey on Core-Sets]**Centers**Core Set points Core-Sets**Core-Sets : Why Care??**• Because they are small ! • Hence we can work on large data sets • Because if we can solve the Optimization problem for Core Sets, we are guaranteed to be near the optimal for the entire input • Because they are easy to compute. • Most algorithms are practical. • Because they exist for even infinite point sets (MEB of balls , ellipsoids, etc)**Core-Sets**Summary of known results for high dimensions.**High Level Algorithm (for most core-set algorithms)**Compute an initial approximation of the solution and its core-set. 2. Find the furthest violator q. 3. Add q to the current core-set and update the corresponding solution. 4. Goto 2 if the solution is not good enough.**Motivation.**• Optimization Formulation. • Initial Volume Approximation. • Algorithm. • Computational Experiments. Axis Aligned Minimum Volume Ellipsoids**Motivation**• Collision Detection [Eberly 01] • Bounding volume Hierarchies • Machine Learning [BJKS 04] • Kernel clustering between MVEs and MEBs?**Optimization Formulation**Volume of unit ball in n-dim space.**Optimization Formulation**Convex**High Level Algorithm (for most core-set algorithms)**Compute an initial approximation of the solution and its core-set. 2. Find the furthest point q from the current solution. 3. Add q to the current core-set and update the corresponding ellipsoid. 4. Goto 2 if the solution is not good enough.**Initial Volume Approximation**Output :**Feasible solution of (LD)**Furthest point from current ellipsoid. Quality measure of current ellipsoid.**Feasible solution of (LD)**Furthest point from current ellipsoid. Quality measure of current ellipsoid.**Increase weight for furthest point**while decreasing it for remaining Points ensuring feasibility for (LD)**What the algorithm outputs?**Complexity:**Computational Experiments**Implementation in Matlab**MVE/AAMVE with outliers**• k-AAMVE Coverings. • Distribution dependent tighter • core-set bounds? • Better practical methods? Future Work Acknowledgements NSF CAREER for support.