A Free Probabilistic Framework for Analyzing the Transformer-based Language Models

TL;DR

This paper introduces a formal operator-theoretic framework using free probability to analyze Transformer-based language models, providing new insights into their spectral dynamics and generalization properties.

Contribution

It develops a novel free probability-based approach to model and analyze the structural and spectral dynamics of Transformers, offering a theoretical perspective.

Findings

Spectral evolution of Transformers can be described by free additive convolution.

Entropy bounds provide insights into model generalization.

Positional encoding impacts spectral and representational complexity.

Abstract

We present a formal operator-theoretic framework for analyzing Transformer-based language models using free probability theory. By modeling token embeddings and attention mechanisms as self-adjoint operators in a tracial \( W^* \)-probability space, we reinterpret attention as non-commutative convolution and describe representation propagation via free additive convolution. This leads to a spectral dynamic system interpretation of deep Transformers. We derive entropy-based generalization bounds under freeness assumptions and provide insight into positional encoding, spectral evolution, and representational complexity. This work offers a principled, though theoretical, perspective on structural dynamics in large language models.