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Geometry of Geometric Data Set I

Shigeaki Yokota

Abstract

Hanika, Schneider, and Stumme introduced geometric data set as a generalization of metric measure space for the computation of the observable diameter, and extended the observable distance between metric measure spaces to that between geometric data sets. In this paper, we begin by proving the non-separability of the observable distance between geometric data sets. We then extend the box distance between mm-spaces to that between geometric data sets and prove its completeness and non-separability.

Geometry of Geometric Data Set I

Abstract

Hanika, Schneider, and Stumme introduced geometric data set as a generalization of metric measure space for the computation of the observable diameter, and extended the observable distance between metric measure spaces to that between geometric data sets. In this paper, we begin by proving the non-separability of the observable distance between geometric data sets. We then extend the box distance between mm-spaces to that between geometric data sets and prove its completeness and non-separability.
Paper Structure (13 sections, 48 theorems, 196 equations)

This paper contains 13 sections, 48 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Metric Measure Space and Coupling
  3. Geometric Data Set
  4. Feature Space and Geometric Data Set
  5. Observable Diameter
  6. Quotient Geometric Data Set
  7. Observable Distance
  8. Definition and Characterization of Observable Distance
  9. The Semi-Continuity of Observable Diameter and the Conversation of Feature Order Relation with respect to the Observable Distance
  10. Box Distance
  11. Definition
  12. Characterization
  13. Completeness

Key Result

Theorem 1.1

The set $\{\ast_A \mid A \subset \mathbb{N}\}$ is an uncountable discrete subset of $(\mathcal{D}, d_{\operatorname{conc}})$. In particular, $(\mathcal{D}, d_{\operatorname{conc}})$ is non-separable.

Theorems & Definitions (117)

  • Theorem 1.1
  • Lemma 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.1: Push-forward
  • Definition 2.2: Support
  • Lemma 2.3: nakajima2022coupling
  • Definition 2.4: Metric measure space, mm-space
  • Remark 2.5
  • Definition 2.6: Prohorov distance
  • ...and 107 more