The task of shape space learning involves mapping a train set of shapes to
and from a latent representation space with good generalization properties.
Often, real-world collections of shapes have symmetries, which can be defined
as transformations that do not change the essence of the shape. A natural way
to incorporate symmetries in shape space learning is to ask that the mapping to
the shape space (encoder) and mapping from the shape space (decoder) are
equivariant to the relevant symmetries.
In this paper, we present a framework for incorporating equivariance in
encoders and decoders by introducing two contributions: (i) adapting the recent
Frame Averaging (FA) framework for building generic, efficient, and maximally
expressive Equivariant autoencoders; and (ii) constructing autoencoders
equivariant to piecewise Euclidean motions applied to different parts of the
shape. To the best of our knowledge, this is the first fully piecewise
Euclidean equivariant autoencoder construction. Training our framework is
simple: it uses standard reconstruction losses and does not require the
introduction of new losses. Our architectures are built of standard (backbone)
architectures with the appropriate frame averaging to make them equivariant.
Testing our framework on both rigid shapes dataset using implicit neural
representations, and articulated shape datasets using mesh-based neural
networks show state-of-the-art generalization to unseen test shapes, improving
relevant baselines by a large margin. In particular, our method demonstrates
significant improvement in generalizing to unseen articulated poses.

本文提出了一种框架，通过引入均变自动编码器和将平面欧几里得运动等价于形状不同部分之间的变换来将均变性纳入编码器和解码器， 在未见过的测试形状上具有最先进的泛化性能。