Deep Polynomial Neural Networks

TL;DR

This paper introduces $ ext{Pi}$-Nets, a new class of polynomial neural networks that use tensor factorization to efficiently model high-order polynomials, achieving state-of-the-art results in various tasks.

Contribution

The paper proposes $ ext{Pi}$-Nets, a novel polynomial neural network architecture utilizing tensor decompositions for efficient high-order polynomial approximation.

Findings

$ ext{Pi}$-Nets are highly expressive even without nonlinear activations.

They achieve state-of-the-art results in image generation, face verification, and 3D mesh learning.

Tensor factorization reduces parameter count significantly.

Abstract

Deep Convolutional Neural Networks (DCNNs) are currently the method of choice both for generative, as well as for discriminative learning in computer vision and machine learning. The success of DCNNs can be attributed to the careful selection of their building blocks (e.g., residual blocks, rectifiers, sophisticated normalization schemes, to mention but a few). In this paper, we propose $\Pi$-Nets, a new class of function approximators based on polynomial expansions. $\Pi$-Nets are polynomial neural networks, i.e., the output is a high-order polynomial of the input. The unknown parameters, which are naturally represented by high-order tensors, are estimated through a collective tensor factorization with factors sharing. We introduce three tensor decompositions that significantly reduce the number of parameters and show how they can be efficiently implemented by hierarchical neural networks. We empirically demonstrate that $\Pi$-Nets are very expressive and they even produce good results without the use of non-linear activation functions in a large battery of tasks and signals, i.e., images, graphs, and audio. When used in conjunction with activation functions, $\Pi$-Nets produce state-of-the-art results in three challenging tasks, i.e. image generation, face verification and 3D mesh representation learning. The source code is available at \url{https://github.com/grigorisg9gr/polynomial_nets}.