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Robust Separation of Multiple Sets.

Lana E Yeganova1, James E Falk1, Yelena V Dandurova1

  • 1707 22nd street, Washington. DC 20052.

Nonlinear Analysis, Theory, Methods & Applications
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PubMed
Summary
This summary is machine-generated.

This study addresses separating finite disjoint sets in Euclidean space using simple functions. It explores linear separability and robust separators when linear separation is possible, or near-separation otherwise.

Keywords:
15A4549N1565K1090C25

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Area of Science:

  • Computational Geometry
  • Machine Learning Theory
  • Optimization

Background:

  • Separating finite disjoint sets is a fundamental problem with broad applications.
  • The existence of separating functions is guaranteed, but linear separation is often preferred.

Purpose of the Study:

  • To investigate methods for finding separating functions for disjoint sets in Euclidean space.
  • To define and explore the concept of piecewise linear separability.
  • To identify robust separators when linear separation is possible and near-separators otherwise.

Main Methods:

  • Exploration of general simple nontrivial functions for set separation.
  • Focus on linear functions for piecewise linear separability.
  • Definition and consideration of robust separators for linearly separable sets.
  • Criteria for near-separation when linear separability is not achieved.

Main Results:

  • General separation is always possible using functions like those derived from Voronoi partitions.
  • Piecewise linear separability is a key consideration when linear functions are sought.
  • Multiple robust separators may exist for linearly separable sets.
  • Approaches for near-separation are considered when sets are not linearly separable.

Conclusions:

  • The study provides a framework for understanding set separation in Euclidean space.
  • It highlights the importance of linear separability and robust separation.
  • It addresses scenarios where perfect linear separation is not attainable, focusing on achieving the closest possible separation.