A free probabilistic framework for analyzing the transformer-based language models

Swagatam Das

Statistics & Probability Letters, 2025, vol. 226, issue C

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

Keywords: Transformers; Free probability; Spectral theory; Non-commutative random variables; Language models (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spl.2025.110516

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