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

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  • Das, Swagatam

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

Suggested Citation

  • Das, Swagatam, 2025. "A free probabilistic framework for analyzing the transformer-based language models," Statistics & Probability Letters, Elsevier, vol. 226(C).
    • Handle: RePEc:eee:stapro:v:226:y:2025:i:c:s0167715225001610
    DOI: 10.1016/j.spl.2025.110516
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    References listed on IDEAS

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    1. Daniel Lehmann, 2009. "Foundations of Non-Commutative Probability Theory," Discussion Paper Series dp514, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    2. Sebastian Farquhar & Jannik Kossen & Lorenz Kuhn & Yarin Gal, 2024. "Detecting hallucinations in large language models using semantic entropy," Nature, Nature, vol. 630(8017), pages 625-630, June.
    3. Benaych-Georges, Florent & Nadakuditi, Raj Rao, 2012. "The singular values and vectors of low rank perturbations of large rectangular random matrices," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 120-135.
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