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General stochastic separation theorems with optimal bounds.

Bogdan Grechuk1, Alexander N Gorban2, Ivan Y Tyukin3

  • 1Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK.

Neural Networks : the Official Journal of the International Neural Network Society
|February 23, 2021
PubMed
Summary
This summary is machine-generated.

Stochastic separability in high-dimensional data enables error correction and vulnerability analysis in Artificial Intelligence (AI). This study provides optimal probability estimates for AI robustness and adaptivity, with implications for neuroscience.

Keywords:
AIAI errorsBlessing of dimensionalityConcentration of measureCurse of dimensionalityDiscriminant

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Area of Science:

  • Machine Learning
  • Artificial Intelligence
  • High-Dimensional Data Analysis
  • Computational Neuroscience

Background:

  • High-dimensional datasets exhibit stochastic separability, where individual points or error clusters can be separated using Fisher's discriminant.
  • This separability is fundamental to understanding Artificial Intelligence (AI) robustness and adaptivity but also introduces vulnerabilities.
  • Existing methods lack precise probability estimates for Fisher separability in high dimensions.

Purpose of the Study:

  • To develop general stochastic separation theorems with optimal probability estimates.
  • To provide tools for managing AI errors and analyzing vulnerabilities in high-dimensional systems.
  • To explore applications in neuroscience, including memory emergence and neural coding.

Main Methods:

  • Derivation of general stochastic separation theorems.
  • Calculation of explicit and optimal probability estimates for Fisher separability.
  • Relaxation of standard independent and identically distributed (i.i.d.) assumptions for distributions.

Main Results:

  • General stochastic separation theorems with optimal probability estimates were obtained for log-concave distributions, their convex combinations, and product distributions.
  • The theorems provide a quantitative understanding of Fisher separability probabilities in high dimensions.
  • The standard i.i.d. assumption was significantly relaxed, broadening applicability.

Conclusions:

  • The derived theorems and estimates are crucial for correcting AI errors and analyzing vulnerabilities in high-dimensional data.
  • These findings offer insights into AI robustness, adaptivity, and potential attack vectors.
  • Applications extend to explaining neural phenomena like sparse coding and memory formation in the brain.