Local Urysohn Width: A Topological Complexity Measure for Classification

2026-03-16

AI Summary

This work proposes local Urysohn width as an intrinsic measure of the topological-geometric complexity of classification problems, characterizing the minimal number of diameter-constrained local experts required to guarantee correct classification. By integrating tools from algebraic topology (e.g., Betti numbers), metric geometry, and statistical learning theory, the paper establishes a rigorous hierarchy theorem and a topological-geometric scaling law, and demonstrates that this measure is bidirectionally separated from the VC dimension. The central contribution lies in revealing the fundamental constraint imposed by Urysohn width on classifier complexity and deriving a sample complexity lower bound of Ω(w log w) that is independent of the VC dimension.

Technology Category

• Machine Learning: Learning Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 $β_1 = w$) 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 $Ω(w \cdot L/D_0)$, 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 $Ω(w \log w)$ samples, independent of VC dimension.

Problem

• Urysohn width
• topological complexity
• classification
• metric space
• sample complexity

Innovation

• local Urysohn width
• topological complexity
• classification
• VC dimension
• sample complexity