On the Stable Euclidean Distance Degree of Algebraic Layers
Giacomo Graziani
TL;DR
The paper investigates the generic Euclidean Distance degree ($gED$) of algebraic neural layers defined by fixed-width polynomial activations, showing that for large input/output dimensions the invariant is a stable polynomial whose form depends only on the activation degree rather than the specific monomial support. The authors unify the invariant with intersection theory on the Nash blow-up, expressing $gED$ as an integral over products of Grassmannians, and prove stability through equivariant localization. A key technical contribution is a reduction to monomial activations of the same degree, achieved via Schubert calculus and uniform bounds on Schubert classes. Altogether, the work provides a structural, dimension-asymptotic understanding of the algebraic complexity of shallow polynomial neural layers and yields practical simplifications for computing $gED$ in large-scale settings.
Abstract
We study the projective geometry of algebraic neural layers, namely families of maps induced by a polynomial activation function, with particular emphasis on the generic Euclidean Distance degree ($\mathrm{gED}$). This invariant is projective in nature and measures the number of optimal approximations of a general point in the ambient space with respect to a general metric. For a fixed architecture (i.e. fixed width and activation polynomial), we prove that the $\mathrm{gED}$ is stably polynomial in the dimensions of the input and output spaces. Moreover, we show that this stable polynomial depends only on the degree of the activation function. Our approach relies on standard intersection theory on the Nash blow-up, which allows us to express the $\gED$ as an intersection number over products of Grassmannians. Stable polynomiality is deduced via equivariant localization, while the reduction to the monomial case follows from an explicit Schubert calculus computation on Grassmannians.