General stochastic separation theorems with optimal bounds

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.