[2605.02907] On the Invariants of Softmax Attention
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Computer Science > Machine Learning
arXiv:2605.02907 (cs)
[Submitted on 6 Apr 2026]
Title:On the Invariants of Softmax Attention
Authors:Wonsuk Lee
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Abstract:Softmax attention maps every query--key interaction into a probability distribution, but the underlying structure remains largely unexplored. We define the \emph{energy field}, the row-centered attention logit, and show that it exhibits invariant properties across models, architectures, and inputs.
Two classes of invariants emerge. \emph{Mechanism-level} invariants follow from the algebraic structure of softmax attention. They include a per-row zero-sum constraint, a rank bound determined by the head dimension, and spectral signatures that follow from them. \emph{Model-level} regularities are not required by the mechanism, yet hold in every autoregressive language model we test, spanning several architecture families. The energy field distributes its variance over key positions without concentrating at a few. This delocalization traces to a property of the key matrix we call \emph{key incoherence}.
These invariants have practical consequences. The rank bound confines the energy field to a low-dimensional subspace. Key incoherence yields a per-head training monitor. All results are verified at multiple context lengths and input texts.
Comments: 15 pages, 3 figures
Subjects:
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
MSC classes: 68T01
ACM classes: I.2.0
Cite as: arXiv:2605.02907 [cs.LG]
(or arXiv:2605.02907v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.02907
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arXiv-issued DOI via DataCite
Submission history
From: Wonsuk Lee [view email] [v1] Mon, 6 Apr 2026 03:21:55 UTC (3,940 KB)
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