Separating balls with partly random hyperplanes with a view to partly random neural networks

TL;DR

This paper derives exact probabilities for separating two Euclidean balls with partly random hyperplanes, highlighting differences from fully random hyperplanes and motivating the study of partly random neural networks.

Contribution

It provides the first exact expressions for separation probabilities with partly random hyperplanes, advancing understanding of partly random neural network models.

Findings

Partly random hyperplanes have higher separation probabilities than fully random ones in certain cases.

Exact formulas for separation probabilities are derived.

Results suggest potential advantages of partly random neural networks.

Abstract

We derive exact expressions for the probabilities that partly random hyperplanes separate two Euclidean balls. The probability that a fully random hyperplane separates two balls turns out to be significantly smaller than the corresponding probabilities for hyperplanes which are not fully random in certain cases. Our results motivate studying partially random neural networks and provide a first step in this direction.