Local Urysohn Width: A Topological Complexity Measure for Classification

TL;DR

This paper introduces local Urysohn width, a topological-geometric complexity measure for classification problems, demonstrating its hierarchical nature, scaling laws, and distinctness from VC dimension, with implications for sample complexity.

Contribution

The paper defines local Urysohn width as a new complexity measure for classification problems, establishing its hierarchy, scaling behavior, and differences from VC dimension.

Findings

Hierarchy theorem linking topological complexity to classifier complexity

Scaling law relating width to topology and geometry

Sample complexity lower bound based on width

Abstract

We introduce \emph{local Urysohn width}, a complexity measure for classification problems on metric spaces. Unlike VC dimension, fat-shattering dimension, and Rademacher complexity, which characterize the richness of hypothesis \emph{classes}, Urysohn width characterizes the topological-geometric complexity of the classification \emph{problem itself}: the minimum number of connected, diameter-bounded local experts needed to correctly classify all points within a margin-safe region. We prove four main results. First, a \textbf{strict hierarchy theorem}: for every integer $w \geq 1$, there exists a classification problem on a \emph{connected} compact metric space (a bouquet of circles with first Betti number $\beta_1 = w$) whose Urysohn width is exactly~$w$, establishing that topological complexity of the input space forces classifier complexity. Second, a \textbf{topology $\times$ geometry scaling law}: width scales as $\Omega(w \cdot L/D_0)$, where $w$ counts independent loops and $L/D_0$ is the ratio of loop circumference to locality scale. Third, a \textbf{two-way separation from VC dimension}: there exist problem families where width grows unboundedly while VC dimension is bounded by a constant, and conversely, families where VC dimension grows unboundedly while width remains~1. Fourth, a \textbf{sample complexity lower bound}: any learner that must correctly classify all points in the safe region of a width-$w$ problem needs $\Omega(w \log w)$ samples, independent of VC dimension.