The Alexander-Hirschowitz theorem for neurovarieties

TL;DR

This paper investigates the geometric properties of neurovarieties in polynomial neural networks, providing a complete characterization of their dimension and implications for model identifiability and non-defectiveness.

Contribution

It offers a comprehensive characterization of neurovarieties' dimensions and establishes conditions for non-defectiveness and identifiability in neural network architectures.

Findings

Neurovarieties attain the expected dimension under certain conditions.

Multi-output architectures are shown to be globally identifiable.

The study confirms non-defectiveness of these varieties.

Abstract

We study neurovarieties for polynomial neural networks and fully characterize when they attain the expected dimension in the single-output case. As consequences, we establish non-defectiveness and global identifiability for multi-output architectures.