Intrinsic Interaction Geometry Controls the Low-Rank Complexity of Softmax Attention
How much matrix rank is required to preserve every bounded value output of normalized softmax attention? We study the unrestricted maximum-row-\(\ell_1\) approximation rank \(r_\varepsilon(A)\), exactly the least rank achieving uniform error over all bounded vector-valued values. Row softmax exposes the intrinsic interaction \(C=P_m(\log A)P_N\), whereas invertible \(Q/K\) gauges leave \(A\) fixed while changing the Euclidean geometry of a chosen query/key factorization. We replace that coordinate-dependent description by a projective residual \(q(C-T)\) and an attained factor-radius size \(κ(T)\). For every rank-\(r\) retained interaction with \(τ(T)<\varepsilon\), we prove $$ r_\varepsi...
| Authors | Yuhe Sui, Jianing Zhang, Yingzhi Tang |
|---|---|
| Primary category | cs.LG |
| Published | 28 August 2026 |
| Last updated at source | 22 September 2026 |
Silytics shows public metadata from arXiv and links to the original. Information may be incomplete or out of date. Last refreshed 17 October 2026. Licenses and terms of the original apply to the work itself. Not financial, legal or professional advice.