Equivariance of linear neural network layers is well studied. In this work, we relax the equivariance condition to only be true in a projective sense. We propose a way to construct a projectively equivariant neural network through building a standard equivariant network where the linear group representations acting on each intermediate feature space are "multiplicatively modified lifts" of projective group representations. By theoretically studying the relation of projectively and linearly equivariant linear layers, we show that our approach is the most general possible when building a network out of linear layers. The theory is showcased in two simple experiments.

翻译：线性神经网络层的等变性已得到充分研究。在本文中，我们将等变性条件松弛为仅在射影意义下成立。我们提出了一种通过构建标准等变网络来构造射影等变神经网络的方法，其中作用于每个中间特征空间的线性群表示是射影群表示的“乘法修正提升”。通过理论研究射影等变线性层与线性等变线性层之间的关系，我们证明了当利用线性层构建网络时，我们的方法是普适性最强的。该理论通过两个简单实验得到了展示。