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Min-Max-Jump distance and its applications

Gangli Liu

TL;DR

The paper introduces Min-Max-Jump distance (MMJ distance), a context-sensitive metric defined as the minimum, over all non-looping paths within a given context, of the maximum jump along the path. It presents three computation strategies—recursion, estimation with copy, and calculation with copy—and demonstrates MMJ’s utility across clustering and labeling tasks, including MMJ-K-means, MMJ-SC, and CNNI enhanced by MMJ-SC, as well as a MMJ-based label-prediction method. Key contributions include a formal MMJ framework, three practical algorithms with varying accuracy and complexity, and a suite of applications showing robust performance on non-spherical and non-convex data, with extensions to directed graphs and widest-path problems. Overall, MMJ distance provides a versatile, context-aware tool for clustering, evaluation, and predictive labeling, with potential for broad applicability in complex data analysis scenarios.

Abstract

We explore three applications of Min-Max-Jump distance (MMJ distance). MMJ-based K-means revises K-means with MMJ distance. MMJ-based Silhouette coefficient revises Silhouette coefficient with MMJ distance. We also tested the Clustering with Neural Network and Index (CNNI) model with MMJ-based Silhouette coefficient. In the last application, we tested using Min-Max-Jump distance for predicting labels of new points, after a clustering analysis of data. Result shows Min-Max-Jump distance achieves good performances in all the three proposed applications. In addition, we devise several algorithms for calculating or estimating the distance.

Min-Max-Jump distance and its applications

TL;DR

The paper introduces Min-Max-Jump distance (MMJ distance), a context-sensitive metric defined as the minimum, over all non-looping paths within a given context, of the maximum jump along the path. It presents three computation strategies—recursion, estimation with copy, and calculation with copy—and demonstrates MMJ’s utility across clustering and labeling tasks, including MMJ-K-means, MMJ-SC, and CNNI enhanced by MMJ-SC, as well as a MMJ-based label-prediction method. Key contributions include a formal MMJ framework, three practical algorithms with varying accuracy and complexity, and a suite of applications showing robust performance on non-spherical and non-convex data, with extensions to directed graphs and widest-path problems. Overall, MMJ distance provides a versatile, context-aware tool for clustering, evaluation, and predictive labeling, with potential for broad applicability in complex data analysis scenarios.

Abstract

We explore three applications of Min-Max-Jump distance (MMJ distance). MMJ-based K-means revises K-means with MMJ distance. MMJ-based Silhouette coefficient revises Silhouette coefficient with MMJ distance. We also tested the Clustering with Neural Network and Index (CNNI) model with MMJ-based Silhouette coefficient. In the last application, we tested using Min-Max-Jump distance for predicting labels of new points, after a clustering analysis of data. Result shows Min-Max-Jump distance achieves good performances in all the three proposed applications. In addition, we devise several algorithms for calculating or estimating the distance.
Paper Structure (22 sections, 6 theorems, 30 equations, 9 figures, 2 tables, 4 algorithms)

This paper contains 22 sections, 6 theorems, 30 equations, 9 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. RELATED WORK
  3. Definition of Min-Max-Jump
  4. An example
  5. As a metric space
  6. Other properties of MMJ distance
  7. Calculation of Min-Max-Jump
  8. MMJ distance by recursion
  9. MMJ distance by estimation and copy
  10. MMJ distance by calculation and copy
  11. Applications of Min-Max-Jump
  12. MMJ-based K-means
  13. MMJ-based Silhouette coefficient
  14. Using MMJ-SC in CNNI
  15. MMJ classifier
  16. ...and 7 more sections

Key Result

Theorem 3.2

Suppose $i, j, p, q \in \Omega$ , then,

Figures (9)

  • Figure 1: An example
  • Figure 2: Four sampled paths between two points
  • Figure 3: Standard K-means vs. MMJ-K-means
  • Figure 4: Eight more samples of MMJ-K-means
  • Figure 5: Clustering result and decision boundary of data B by CNNI using MMJ-SC
  • ...and 4 more figures

Theorems & Definitions (10)

  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Theorem 3.5
  • Definition 5.1
  • Theorem 6.1
  • Theorem 6.2
  • Definition 8.1
  • Definition 8.2