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
https://github.com/grigorisg9gr/polynomial_nets.

本文提出了一种基于多项式扩展的新型函数逼近器 ——π- 网，它是多项式神经网络，通过具有共享因子的张量集合分解来估计自然表示的未知参数，可用于许多任务和信号的表达建模，并在激活函数的辅助下在图像生成、人脸验证和 3D 网格表示学习等任务中实现了最新的结果。