Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local Urysohn Width: A Topological Complexity Measure for Classification

Xin Li

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.

Local Urysohn Width: A Topological Complexity Measure for Classification

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 , there exists a classification problem on a \emph{connected} compact metric space (a bouquet of circles with first Betti number ) whose Urysohn width is exactly~, establishing that topological complexity of the input space forces classifier complexity. Second, a \textbf{topology geometry scaling law}: width scales as , where counts independent loops and 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- problem needs samples, independent of VC dimension.
Paper Structure (52 sections, 12 theorems, 21 equations, 1 figure)

This paper contains 52 sections, 12 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Related work.
  4. The Urysohn Machine
  5. Informal Overview
  6. Formal Definition
  7. Width as Library Size
  8. Local Urysohn Width
  9. Formal Definitions
  10. Basic Properties
  11. Hierarchy on Connected Spaces via beta-1
  12. Construction: Bouquet of Circles
  13. Main Result
  14. Topology $\times$ Geometry Scaling Law
  15. Construction: Multiple Safe Regions per Loop
  16. ...and 37 more sections

Key Result

Lemma 3.5

If $0<\gamma' < \gamma$, then $\mathrm{uw}_{D_0}(\mathcal{P},\gamma') \le \mathrm{uw}_{D_0}(\mathcal{P},\gamma)$.

Figures (1)

  • Figure 1: Urysohn width captures problem structure that VC dimension does not.(a) A single complex decision boundary requires an expressive classifier ($\mathrm{VC} = \Theta(n)$) but only one local expert ($\mathrm{uw} = 1$): the entire space lies within a single connected patch $S_1$. (b) Five disjoint pairs of trivially separable classes need only constant classifiers ($\mathrm{VC} \leq 1$) but require five separate patches $S_1, \ldots, S_5$, giving $\mathrm{uw} = 5$. (c) The bouquet construction (Theorem \ref{['thm:hierarchy']}): a connected space with first Betti number $\beta_1 = 3$. Each safe region $A_j$ sits on a distinct loop, and the locality constraint ($\mathrm{diam}(S_i) \leq D_0 < L/2 - 3\gamma/4$) confines every patch to a single loop. The topology of the space forces $\mathrm{uw}_{D_0} \geq \beta_1$. Dashed green boxes indicate local Urysohn triples.

Theorems & Definitions (47)

  • Definition 2.1: Urysohn Machine
  • Definition 3.1: Margin partition and safe region
  • Definition 3.2: Local Urysohn triple and $(\gamma, D_0)$-covering
  • Definition 3.3: Local Urysohn width
  • Remark 3.4: Why locality is essential
  • Lemma 3.5: Monotonicity in margin
  • proof
  • Lemma 3.6: Monotonicity under refinement
  • proof
  • Lemma 3.7: Additivity on separated disjoint unions
  • ...and 37 more