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Language Modeling with Reduced Densities

Tai-Danae Bradley, Yiannis Vlassopoulos

TL;DR

The paper investigates the mathematical structure of language in unstructured text and proposes a framework that merges enriched category theory with quantum-inspired density operators. It shows that sequences over a finite alphabet form a category enriched over probabilities and constructs a functor to an enriched category of reduced density operators, with the Loewner order providing a formal notion of entailment. The key contributions include formalizing $[0,1]$-enrichment, deriving unit-trace reduced densities for phrases, and proving that the mapping $s\mapsto \rho_s$ preserves the language preorder, thereby capturing both compositional and statistical aspects of meaning. The approach offers a principled, structure-preserving representation of linguistic semantics that can be approximated with tensor-network methods for scalable language modeling.

Abstract

This work originates from the observation that today's state-of-the-art statistical language models are impressive not only for their performance, but also - and quite crucially - because they are built entirely from correlations in unstructured text data. The latter observation prompts a fundamental question that lies at the heart of this paper: What mathematical structure exists in unstructured text data? We put forth enriched category theory as a natural answer. We show that sequences of symbols from a finite alphabet, such as those found in a corpus of text, form a category enriched over probabilities. We then address a second fundamental question: How can this information be stored and modeled in a way that preserves the categorical structure? We answer this by constructing a functor from our enriched category of text to a particular enriched category of reduced density operators. The latter leverages the Loewner order on positive semidefinite operators, which can further be interpreted as a toy example of entailment.

Language Modeling with Reduced Densities

TL;DR

The paper investigates the mathematical structure of language in unstructured text and proposes a framework that merges enriched category theory with quantum-inspired density operators. It shows that sequences over a finite alphabet form a category enriched over probabilities and constructs a functor to an enriched category of reduced density operators, with the Loewner order providing a formal notion of entailment. The key contributions include formalizing -enrichment, deriving unit-trace reduced densities for phrases, and proving that the mapping preserves the language preorder, thereby capturing both compositional and statistical aspects of meaning. The approach offers a principled, structure-preserving representation of linguistic semantics that can be approximated with tensor-network methods for scalable language modeling.

Abstract

This work originates from the observation that today's state-of-the-art statistical language models are impressive not only for their performance, but also - and quite crucially - because they are built entirely from correlations in unstructured text data. The latter observation prompts a fundamental question that lies at the heart of this paper: What mathematical structure exists in unstructured text data? We put forth enriched category theory as a natural answer. We show that sequences of symbols from a finite alphabet, such as those found in a corpus of text, form a category enriched over probabilities. We then address a second fundamental question: How can this information be stored and modeled in a way that preserves the categorical structure? We answer this by constructing a functor from our enriched category of text to a particular enriched category of reduced density operators. The latter leverages the Loewner order on positive semidefinite operators, which can further be interpreted as a toy example of entailment.

Paper Structure

This paper contains 11 sections, 10 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Acknowledgments
  4. Modeling Probability Distributions with Density Operators
  5. Understanding Reduced Densities
  6. Why Densities for Language?
  7. Assigning Reduced Densities to Words
  8. Preserving a Preorder Structure
  9. Language as a Preordered Set
  10. Language as a Category Enriched Over Probabilities
  11. Conclusion

Key Result

Proposition 2.1

Let $\pi\colon X\times Y\to\mathbb{R}$ be a probability distribution and let $\psi$ be the vector given in Equation eq:psi2. Suffixes $y_c$ and $y_d$ satisfy $\pi(x_i,y_c)=\pi(x_i,y_d)$ for all $i$ if and only if they have the same image under $\rho_Y=\mathop{\mathrm{tr}}\nolimits_X\text{Pr}_\psi$.

Figures (3)

  • Figure 1: A tensor network diagram illustrating the construction of the reduced densities $\rho_V$ and $\hat{\rho}_{x_{i_2}x_{i_1}}$ from the unit vector $\psi.$
  • Figure 2: A rank 1 operator illustrated as a tensor network diagram with no contracted edges.
  • Figure 3: Associating reduced densities to phrases while leaving the first and last indices open.

Theorems & Definitions (26)

  • Proposition 2.1
  • proof
  • Example 1
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Example 2
  • Proposition 3.1
  • proof
  • Example 3
  • ...and 16 more