Core-Sets and Geometric Optimization problems.

Piyush KumarDepartment of Computer Science

Florida State University

http://piyush.compgeom.comEmail: piyush@acm.org

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

CentersCore Set points

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)

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

Optimization Formulation

Volume of unit ball in n-dim space.

Optimization Formulation

Con

vex

Optimization Formulation

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.

Optimization Formulation: Lemma 1

Initial Volume Approximation

Output :

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)

What the algorithm outputs?

Complexity:

Computational Experiments

Implementation in Matlab

Future Work

o MVE/AAMVE with outlierso k-AAMVE Coverings.o Distribution dependent tighter core-set bounds?o Better practical methods?

AcknowledgementsNSF CAREER for support.