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Monotonicity of linear separability under translation.

A M Bruckstein1, T M Cover

  • 1Department of Electrical Engineering, Stanford University, Stanford, CA 94305.

IEEE Transactions on Pattern Analysis and Machine Intelligence
|August 27, 2011
PubMed
Summary

Translating pattern vector sets can improve linear separability, especially on average. The probability of separating two data sets with a hyperplane increases as the translation distance grows.

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Area of Science:

  • Machine Learning
  • Pattern Recognition
  • Computational Geometry

Background:

  • Linear separability is a fundamental concept in pattern recognition.
  • A hyperplane can separate two sets of points if they are linearly separable.
  • The effect of data translation on separability is not fully understood.

Purpose of the Study:

  • To investigate if translating one set of pattern vectors enhances linear separability.
  • To determine the average effect of arbitrary translations on separability.
  • To analyze the probability of linear separability as a function of translation distance.

Main Methods:

  • Considered a set of n pattern vectors in d-space, arbitrarily classified into two sets.
  • Defined linear separability by the existence of a separating hyperplane.
  • Analyzed the probability of separability over all possible classifications and arbitrary translations.

Main Results:

  • Translation of one set can sometimes improve, and sometimes hinder, linear separability.
  • On average, translation increases the likelihood of linear separability.
  • The probability of linear separability was proven to increase with translation distance (t).

Conclusions:

  • Linear separability is generally enhanced by translating one of the pattern sets.
  • The probability of linear separability is minimized at zero translation (t=0) and increases for t>0.
  • This finding has implications for data distribution and classification algorithms.