The Alexander-Hirschowitz theorem for neurovarieties

Abstract

We study the dimension and identifiability of neurovarieties associated to polynomial neural networks. We give an independent geometric proof that the linear bounds $d_i\ge 2n_i-1$ on the activation degrees imply non defectiveness for any number of outputs, a dimension statement previously obtained from finite identifiability. The proof is based on a direct analysis of the differential of the parameterization. We also investigate secant and Grassmann-secant obstructions outside this range and prove global identifiability for multi-output architectures under the same degree bounds.

15 pages. Thanks to Kevin Dao, we identified a gap in the previous proof of the main non-defectiveness result. We replaced the transversality argument with a direct analysis of the differential, based on a new linear independence lemma, obtaining the degree bound $d_i\ge 2n_i-1$