Minimal Filling Architectures of Polynomial Neural Networks: Counterexamples, Frontier Search, and Defects

We study the expressivity of shallow polynomial neural networks (PNNs) with monomial activation functions over finite fields. For a given architecture, we define a neuromanifold as the image of the map from all possible network weights into the product of polynomial rings. We quantify the expressivity by the cardinality of the neuromanifold, and derive a natural lower and upper bound. This leads to counting rational points over finite fields, a problem closely linked to the Weil conjectures. Finally, we present an architecture that exhibits a striking difference in the neuromanifolds when considered over a characteristic zero versus a finite-characteristic field, illustrating the critical role of field characteristic in the notion of expressivity.

We study the dimension and identifiability of neurovarieties associated to polynomial neural networks. We give an independent geometric proof that the linear bounds $d_i\geq 2n_i-1$ 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.

In spite of the fundamental role of neural networks in contemporary machine learning research, our understanding of the computational complexity of optimally training neural networks remains incomplete even when dealing with the simplest kinds of activation functions. Indeed, while there has been a number of very recent results that establish ever-tighter lower bounds for the problem under linear and ReLU activation functions, less progress has been made towards the identification of novel polynomial-time tractable network architectures. In this article we obtain novel algorithmic upper bounds for training linear- and ReLU-activated neural networks to optimality which push the boundaries of tractability for these problems beyond the previous state of the art. In particular, for ReLU networks we establish the polynomial-time tractability of all architectures where hidden neurons have an out-degree of $1$, improving upon the previous algorithm of Arora, Basu, Mianjy and Mukherjee. On the other hand, for networks with linear activation functions we identify the first non-trivial polynomial-time solvable class of networks by obtaining an algorithm that can optimally train network architectures satisfying a novel data throughput condition.