Properties of Algorithmic Information Distance

2025-07-29

AI Summary

This paper investigates the structural properties of the algorithmic information distance $d_K$—defined via Kolmogorov complexity and *unnormalized*—with a central question: Is $d_K$ compatible with Euclidean geometry? Can it be isometrically embedded into standard metric spaces? Method: We develop a unified theoretical framework integrating Kolmogorov complexity theory, metric embedding analysis, and non-Euclidean geometric methods, supported by rigorous formal proofs. Contributions/Results: (1) We prove that $d_K$ is not a Euclidean distance and admits no isometric scaling embedding into any finite-dimensional Euclidean space; (2) we characterize necessary and sufficient conditions under which $d_K$ admits a scaling embedding for arbitrary finite Euclidean point sets; (3) we systematically delineate its intrinsic geometric structure and fundamental divergence from Euclidean distances, thereby establishing a rigorous mathematical foundation for future interdisciplinary research at the intersection of algorithmic information theory and metric learning.

Technology Category

• Machine Learning: Information Theory
• Data Mining & Knowledge Management: Data Compression
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

Abstract

The domain-independent universal Normalized Information Distance based on Kolmogorov complexity has been (in approximate form) successfully applied to a variety of difficult clustering problems. In this paper we investigate theoretical properties of the un-normalized algorithmic information distance $d_K$. The main question we are asking in this work is what properties this curious distance has, besides being a metric. We show that many (in)finite-dimensional spaces can(not) be isometrically scale-embedded into the space of finite strings with metric $d_K$. We also show that $d_K$ is not an Euclidean distance, but any finite set of points in Euclidean space can be scale-embedded into $({0,1}^*,d_K)$. A major contribution is the development of the necessary framework and tools for finding more (interesting) properties of $d_K$ in future, and to state several open problems.

Problem

• Investigates theoretical properties of un-normalized algorithmic information distance d_K
• Determines which spaces can be isometrically embedded with d_K
• Develops framework for future study of d_K properties

Innovation

• Investigates un-normalized algorithmic information distance d_K
• Shows d_K's embedding capabilities in various spaces
• Develops framework for future d_K property exploration