CP-MLPs: A Tensor-Rank Theory of Tied and Untied MLP Blocks

Transformer feed-forward blocks combine several design choices: squared activations, gating, and tied versus untied parameter sharing, whose separate contributions to expressivity remain poorly understood. We introduce CP-MLPs, a tensor framework in which each multiplicative unit contributes a rank-1 interaction with three roles: detecting an input feature, reading a value, and writing to the output. Tied gated blocks such as ReLU^2 force the detector and value roles to share the same direction; untied gated blocks such as ReGLU and SwiGLU let them differ. We show that this distinction yields exact, degree-wise tensor-rank separations: untied detector-value units are strictly more width-efficient for indefinite quadratic interactions and role-asymmetric monomials, while tied units are structurally matched to symmetric pure-power targets. The width advantage of untying therefore depends on the target structure. Experiments show loss plateaus below the exact-representation width, but these plateaus alone do not establish a positive approximation gap: arbitrarily accurate approximation is governed by border rank. In trained gated MLPs, untied blocks retain an advantage in selected comparisons at matched input-projection budget. Code is available at https://github.com/rarefin/CP-MLPs.