Year: 2026
Type: preprint
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 $2^{Θ(m)}$ 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 $2^{Θ_H(m)}$ 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 $Ω(n\log m)$ bits over a range of lengths that grows exponentially with $m$.
Source: arXiv (Cornell University)
Author: Opazo, Vicente
Topic: Topic Modeling
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202355490
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