A Language and Its Dimensions: Intrinsic Dimensions of Language Fractal Structures

TL;DR

This paper introduces the concept of language fractal structures, estimating their intrinsic dimensions for Russian and English using topological data analysis and minimum spanning trees, revealing non-integer dimensions close to 9.

Contribution

It proposes a novel framework for analyzing language as fractal structures and estimates their intrinsic dimensions, a new approach in linguistic complexity analysis.

Findings

Intrinsic dimensions are non-integer, close to 9 for both languages.

Language fractal structures can be characterized by topological and graph-based methods.

Russian and English share similar fractal dimension properties.

Abstract

The present paper introduces a novel object of study - a language fractal structure. We hypothesize that a set of embeddings of all $n$-grams of a natural language constitutes a representative sample of this fractal set. (We use the term Hailonakea to refer to the sum total of all language fractal structures, over all $n$). The paper estimates intrinsic (genuine) dimensions of language fractal structures for the Russian and English languages. To this end, we employ methods based on (1) topological data analysis and (2) a minimum spanning tree of a data graph for a cloud of points considered (Steele theorem). For both languages, for all $n$, the intrinsic dimensions appear to be non-integer values (typical for fractal sets), close to 9 for both of the Russian and English language.