The Approximation Rank of Softmax Attention: Sharp Geometric Laws and Robust Interaction Dimension
Which geometry controls the rank complexity of normalized softmax attention? We study maximum-row-$\ell_1$ approximation rank, exactly the least unrestricted rank preserving every bounded vector-valued output. Two sharp worst-case laws isolate support geometry: for fixed $d$ and error $\varepsilon$, spherical self-attention has rank $Θ_{d,\varepsilon}(\min\{n,(1+β)^{(d-1)/2}\})$, while full-ball geometry adds one radial degree and, for $β\geβ_0(d,\varepsilon)$ and $n\ge C_d e^{β/8}$, gives $Θ_{d,\varepsilon}(β^{d/2})$. For a fixed head, row-softmax quotients out row-scalar logit directions: the remaining visible query--key interaction dimension $r$ yields an $r/2$ per-instance upper law, and bounded constructions show this exponent is minimax sharp. Approximate interaction subspaces incur an explicit residual output error and yield a tolerance-indexed SVD dimension. On an 84-head BERT-base calibration set, we observe modest effective-dimension reductions across many head--temperature settings, together with positive associations with finite constructive rank upper certificates. Together, these results separate support geometry, which sets worst-case temperature scaling, from softmax-visible interaction geometry, which controls per-head approximation complexity.