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arXiv · 2506.16550

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

Swagatam Das
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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.

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Swagatam Das. 2025-06-19. A Free Probabilistic Framework for Analyzing the Transformer-based Language Models. https://arxiv.org/abs/2506.16550

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