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被广泛研究。本文中，我们将等变性条件放宽至仅满足投影意义上的等变性。我们提出了一种通过构建标准等变网络来构造投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投影群表示的"乘法修正提升"。通过从理论上研究投影等变线性层与线性等变线性层之间的关系，我们证明了该方法在基于线性层构建网络时具有最大普适性。该理论通过两个简单实验进行了展示。