Three Tokens Force Exponential Feature Rank in Nonnegative Kernel Attention

arXiv:2608.11427v2 Announce Type: replace Abstract: How much feature rank does comparison require in kernel attention? On Min-IP over $m$-bit tokens, rank one solves every sequence of length at most two exactly. At length three, the minimum feature rank of one normalized nonnegative kernel-attention head is $2^{\Theta(m)}$ for error strictly below $1/2$ on every input, even with arbitrary finite-dimensional tokenwise values and query-dependent affine readouts. Dense softmax solves this three-token task with $m$-dimensional scores and temperature constant in $m$. For every fixed number of heads $H$, the minimum total feature rank is $2^{\Theta_H(m)}$ for the same error guarantee at exact length $H+2$ in one attention layer with affine mixing. These bounds also hold with position-dependent maps and a final causal query. For one head with polynomial readout of fixed degree at most $D$, rank one suffices at exact length $D+1$, while exact length $D+2$ requires exponential feature rank. With unrestricted exact-real decoding, a scalar rank-one construction solves the task at every finite length. This motivates a separate bound on total communication for deterministic models with finite-alphabet cross-token channels and any number of heads and layers. In this setting, correctness up to length $n$ requires $\Omega(n\log m)$ bits over a range of lengths that grows exponentially with $m$.

Abstract:How much feature rank does comparison require in kernel attention? On Min-IP over $m$-bit tokens, rank one solves every sequence of length at most two exactly. At length three, the minimum feature rank of one normalized nonnegative kernel-attention head is $2^{\Theta(m)}$ for error strictly below $1/2$ on every input, even with arbitrary finite-dimensional tokenwise values and query-dependent affine readouts. Dense softmax solves this three-token task with $m$-dimensional scores and temperature constant in $m$. For every fixed number of heads $H$, the minimum total feature rank is $2^{\Theta_H(m)}$ for the same error guarantee at exact length $H+2$ in one attention layer with affine mixing. These bounds also hold with position-dependent maps and a final causal query. For one head with polynomial readout of fixed degree at most $D$, rank one suffices at exact length $D+1$, while exact length $D+2$ requires exponential feature rank. With unrestricted exact-real decoding, a scalar rank-one construction solves the task at every finite length. This motivates a separate bound on total communication for deterministic models with finite-alphabet cross-token channels and any number of heads and layers. In this setting, correctness up to length $n$ requires $\Omega(n\log m)$ bits over a range of lengths that grows exponentially with $m$.

27 pages, 3 figures. New theorems and experiments

Machine Learning (cs.LG)

arXiv:2608.11427 [cs.LG]

(or arXiv:2608.11427v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.11427 Focus to learn more arXiv-issued DOI via DataCite

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Three Tokens Force Exponential Feature Rank in Nonnegative Kernel Attention

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