core-sets and geometric optimization problems
DESCRIPTION
Core-Sets and Geometric Optimization problems. Piyush Kumar Department of Computer Science Florida State University http://piyush.compgeom.com Email: [email protected]. Joint work with. Alper Yildirim. Talk Outline. Introduction to Core-Sets for Geometric Optimization Problems. - PowerPoint PPT PresentationTRANSCRIPT
Core-Sets and Geometric Optimization problems.
Piyush KumarDepartment of Computer Science
Florida State University
http://piyush.compgeom.comEmail: [email protected]
Alper Yildirim
Joint work with
1. Introduction to Core-Sets for Geometric Optimization
Problems.
2. Problems and Applications
1. Minimum Enclosing Balls (Next talk)
2. Axis Aligned Minimum Volume Ellipsoids
1. Motivation
2. Optimization Formulation/IVA
3. Algorithm
4. Computational Experiments
3. Future Directions.
Talk Outline
In order to cluster, we need
o Points ( Polyhedra / Balls / Ellipsoids ? ) o A distance measure
Assume Euclidian unless otherwise stated. o A method for evaluating our clustering. (We look at k-centers, 1-Ecenter, 1-center, kernel 1-center, k-line-centers)
Geometric Clustering
r*
For some shapes, the problem is convex and hence tractable. (MEB / MVE / AAMVE).
o Minimize the maximum distance O(n) in 2D but becomes harder in higher dimensions.
o Least squares / SVM Regression / …
o d ≠ O(1)?
Fitting/Covering Problems
o Non-Convex (Min the max distance) / Non-Linear / NP-Hard to approx.
o Has many applications
o Minimize sum of distances (orthogonal) : SVD
o Other assumptions : GPCA/PPCA/PCA…
Fitting multiple subspaces/shapes?
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]
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)
High Level Algorithm (for most core-set algorithms)
1. 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.
Axis Aligned Minimum Volume Ellipsoids
o Motivation.o Optimization Formulation.o Initial Volume Approximation.o Algorithm.o Computational Experiments.
o Collision Detection [Eberly 01]o Bounding volume Hierarchies
o Machine Learning [BJKS 04]o Kernel clustering between MVEs and MEBs?
Motivation
High Level Algorithm (for most core-set algorithms)
1. 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.
Feasible solution of (LD)
Furthest point from currentellipsoid.
Quality measure of currentellipsoid.
Feasible solution of (LD)
Furthest point from currentellipsoid.
Quality measure of currentellipsoid.
Increase weight for furthest pointwhile decreasing it for remainingPoints ensuring feasibility for (LD)