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Bounding the errors for convex dynamics on one or more polytopes.

Charles Tresser¹

¹IBM T. J. Watson Research Center, P.O. Box 218, Route 134, Kitchawan Road, Yorktown Heights, New York 10598, USA.

Chaos (Woodbury, N.Y.)

|October 2, 2007

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This study analyzes a greedy algorithm for approximating polytope sequences, focusing on Euclidean error norms. Researchers proved the existence of invariant sets for bounded polytope families, challenging prior conjectures on error saturation.

Area of Science:

Computational Geometry
Optimization Algorithms
Dynamical Systems

Background:

The greedy algorithm approximates input sequences within polytopes using vertex sequences.
Error bounds were previously established for single polytopes using invariant regions in affine spaces.
General cases with multiple polytopes showed no invariant affine space regions.

Purpose of the Study:

To investigate the existence of invariant sets for a greedy algorithm approximating sequences in families of polytopes.
To analyze the behavior of cumulative errors in dynamical systems derived from this algorithm.
To challenge existing conjectures regarding error saturation in polytope approximation.

Main Methods:

Interpreting the greedy algorithm as a time-dependent dynamical system in vector and affine spaces.

Related Experiment Videos

Proving the existence of large convex invariant sets in the vector space for bounded polytope families with finite outgoing normals.

Demonstrating error set saturation issues for specific single polytopes (quadrilaterals, pentagons), contradicting previous conjectures.

Main Results:

Existence of arbitrary large convex invariant sets in the vector space is proven under specific conditions (bounded sizes, finite normals).
Error saturation in finite steps is shown to occur even for single quadrilaterals and axially symmetric pentagons.
This disproves a former conjecture and suggests a potential pathway for proving bounded errors in general polytope families.

Conclusions:

The study establishes conditions for invariant set existence in vector spaces for greedy polytope approximation.
A key conjecture regarding error saturation is disproven, highlighting complexities in multi-polytope systems.
Findings suggest that bounding errors in general polytope families may be achievable with further theoretical development.