On the probability of linear separability through intrinsic volumes

TL;DR

This paper derives a formula and bounds for the probability of linear separability in Gaussian datasets, linking geometric properties of polyhedral cones to statistical questions about data separability.

Contribution

It introduces a geometric approach using intrinsic volumes of polyhedral cones to compute the probability of linear separability, extending recent theoretical results.

Findings

Provides a formula for Gaussian feature data

Derives an upper bound complementing recent work

Calculates intrinsic volumes using a new projection algorithm

Abstract

A dataset with two labels is linearly separable if it can be split into its two classes with a hyperplane. This inflicts a curse on some statistical tools (such as logistic regression) but forms a blessing for others (e.g. support vector machines). Recently, the following question has regained interest: What is the probability that the data are linearly separable? We provide a formula for the probability of linear separability for Gaussian features and labels depending only on one marginal of the features (as in generalized linear models). In this setting, we derive an upper bound that complements the recent result by Hayakawa, Lyons, and Oberhauser [2023], and a sharp upper bound for sign-flip noise. To prove our results, we exploit that this probability can be expressed as a sum of the intrinsic volumes of a polyhedral cone of the form $\text{span}\{v\}\oplus[0,\infty)^n$, as shown in Cand\`es and Sur [2020]. After providing the inequality description for this cone, and an algorithm to project onto it, we calculate its intrinsic volumes. In doing so, we encounter Youden's demon problem, for which we provide a formula following Kabluchko and Zaporozhets [2020]. The key insight of this work is the following: The number of correctly labeled observations in the data affects the structure of this polyhedral cone, allowing the translation of insights from geometry into statistics.