On the Expressive Power of Deep Polynomial Neural Networks
We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines an algebraic map from weights to polynomials. The image of this map is the functional space associated to the network, and it is an irreducible algebraic variety upon taking closure. This paper proposes the dimension of this variety as a precise measure of the expressive power of polynomial neural networks. We obtain several theoretical results regarding this dimension as a function of architecture, including an exact formula for high activation degrees, as well as upper and lower bounds on layer widths in order for deep polynomials networks to fill the ambient functional space. We also present computational evidence that it is profitable in terms of expressiveness for layer widths to increase monotonically and then decrease monotonically. Finally, we link our study to favorable optimization properties when training weights, and we draw intriguing connections with tensor and polynomial decompositions.
Code (1)
Tasks
Polynomial Neural NetworksSimilar Papers 제목 키워드 기반
Equivariant Polynomials for Graph Neural Networks
Graph Neural Networks (GNN) are inherently limited in their expressive power. Recent seminal works (Xu et al., 2019; Morris et al., 2019b) introduced the Weisfeiler-Lehman (WL) hierarchy as a measure of expressive power.…
Graph LearningActivation degree thresholds and expressiveness of polynomial neural networks
We study the expressive power of deep polynomial neural networks through the geometry of their neurovariety. We introduce the notion of the activation degree threshold of a network architecture to express when the dimens…
Polynomial Neural NetworksImproving Expressive Power of Spectral Graph Neural Networks with Eigenvalue Correction
In recent years, spectral graph neural networks, characterized by polynomial filters, have garnered increasing attention and have achieved remarkable performance in tasks such as node classification. These models typical…
Node ClassificationDepth separation and weight-width trade-offs for sigmoidal neural networks
Some recent work has shown separation between the expressive power of depth-2 and depth-3 neural networks. These separation results are shown by constructing functions and input distributions, so that the function is wel…
Graphical Representations for Algebraic Constraints of Linear Structural Equations Models
The observational characteristics of a linear structural equation model can be effectively described by polynomial constraints on the observed covariance matrix. However, these polynomials can be exponentially large, mak…