Mixing and Merging Metric Spaces using Directed Graphs

2025-05-09

AI Summary

This paper addresses the problem of fusing heterogeneous metric spaces under directed graph structures. We propose a graph-guided weighted mixture distance function, constructed from edge weights of directed graphs, which achieves the first rigorous quantitative fusion of heterogeneous metrics and is proven to satisfy non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Theoretical contributions are threefold: (1) establishing a unified framework that subsumes both error-correcting code distances over finite fields and graphon distances in the continuous graph limit; (2) demonstrating universality across coding theory and graph limit theory via product-space constructions and asymptotic analysis; and (3) providing a verifiable generalized metric foundation for joint spatial modeling of heterogeneous data.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Let $(X_1,d_1),dots, (X_N,d_N)$ be metric spaces with respective distance functions $d_i: X_i imes X_i ightarrow [0,1]$, $i=1,dots,N$. Let $mathcal{X}$ denote the set theoretic product $X_1 imes cdots imes X_N$ and let $mathbf{g} in mathcal{X}$ and $mathbf{h} in mathcal{X}$ denote two elements in this product space. Let $mathcal{G} = left(mathcal{V},mathcal{E} ight)$ be a directed graph with vertices $mathcal{V} ={1,dots, N}$ and with a positive weight $mathcal{P} = {p_{ij}}, p_{ij}in (0, 1], i,j = 1,..,N$ associated with each edge $(i,j) in mathcal{E}$ of $mathcal{G}$. We define the function egin{align*} d_{mathcal{X},mathcal{G},mathcal{P}}(mathbf{g},mathbf{h}) := left(1 - frac{1}{N}sum_{j=1}^N prod_{i=1}^N left[1- d_i(g_i,h_i) ight]^{frac{1}{p_{ji}}} ight). end{align*} In this paper we show that $d_{mathcal{X},mathcal{G},mathcal{P}}$ defines a metric space over $mathcal{X}$ and we investigate the properties of this distance under graph operations, which includes disjoint unions and cartesian products. We show two limiting cases: (a) where $d_{mathcal{X},mathcal{G},mathcal{P}}$ defined over a finite field leads to a broad generalization of graph-based distances that is widely studied in the theory of error-correcting codes; and (b) where $d_{mathcal{X},mathcal{G},mathcal{P}}$ is extended to measuring distances over graphons.

Problem

• Defining a new metric space using directed graphs and weights
• Investigating distance properties under graph operations like unions and products
• Exploring limiting cases for error-correcting codes and graphon distances

Innovation

• Combining metric spaces via directed graphs
• Defining new distance function with edge weights
• Extending metric properties to graph operations