  In this paper, we revisit the classical representation of 3D point clouds as
linear shape models. Our key insight is to leverage deep learning to represent
a collection of shapes as affine transformations of low-dimensional linear
shape models. Each linear model is characterized by a shape prototype, a
low-dimensional shape basis and two neural networks. The networks take as input
a point cloud and predict the coordinates of a shape in the linear basis and
the affine transformation which best approximate the input. Both linear models
and neural networks are learned end-to-end using a single reconstruction loss.
The main advantage of our approach is that, in contrast to many recent deep
approaches which learn feature-based complex shape representations, our model
is explicit and every operation occurs in 3D space. As a result, our linear
shape models can be easily visualized and annotated, and failure cases can be
visually understood. While our main goal is to introduce a compact and
interpretable representation of shape collections, we show it leads to state of
the art results for few-shot segmentation.
