Language Modeling with Reduced Densities
This work addresses a foundational problem in language modeling by introducing a novel theoretical framework, but it appears incremental as it builds on existing mathematical concepts without clear practical application.
The paper tackles the problem of understanding the mathematical structure in unstructured text data by proposing enriched category theory as a framework, and it constructs a functor to model this structure using reduced density operators, though no concrete performance numbers are provided.
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.