1. Core-Sets and Geometric Optimization problems.
Piyush Kumar
Department of Computer Science
Florida State University
Joint work with
Alper Yildirim

2. 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.

3. In order to cluster, we need
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)

4. 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)?

5. 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…

6. [AHV06 Survey on 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]

7. Centers
Core Set points
Core-Sets

8. 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)

9. Core-Sets
Summary of known results for high dimensions.

10. 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.

11. Axis Aligned Minimum Volume Ellipsoids
Motivation.
Optimization Formulation.
Initial Volume Approximation.
Algorithm.
Computational Experiments.

12. Motivation
Collision Detection [Eberly 01]
Bounding volume Hierarchies
Machine Learning [BJKS 04]
Kernel clustering between MVEs and MEBs?

13. Optimization Formulation
Volume of unit ball in n-dim space.

14. Optimization Formulation
Convex

15. Optimization Formulation

16. 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.

17. Optimization Formulation: Lemma 1

18. Initial Volume Approximation
Output :

19. Feasible solution of (LD)
Furthest point from current
ellipsoid.
Quality measure of current
ellipsoid.

20. Feasible solution of (LD)
Furthest point from current
ellipsoid.
Quality measure of current
ellipsoid.

21. Increase weight for furthest point
while decreasing it for remaining
Points ensuring feasibility for (LD)

22. What the algorithm outputs?
Complexity:

23. Computational Experiments
Implementation in Matlab

27. MVE/AAMVE with outliers k-AAMVE Coverings.
Future Work
MVE/AAMVE with outliers
k-AAMVE Coverings.
Distribution dependent tighter
core-set bounds?
Better practical methods?
Acknowledgements
NSF CAREER for support.