  In this paper, we propose NeuralQAAD, a differentiable point cloud
compression framework that is fast, robust to sampling, and applicable to high
resolutions. Previous work that is able to handle complex and non-smooth
topologies is hardly scaleable to more than just a few thousand points. We
tackle the task with a novel neural network architecture characterized by
weight sharing and autodecoding. Our architecture uses parameters much more
efficiently than previous work, allowing us to be deeper and scalable.
Futhermore, we show that the currently only tractable training criterion for
point cloud compression, the Chamfer distance, performances poorly for high
resolutions. To overcome this issue, we pair our architecture with a new
training procedure based upon a quadratic assignment problem (QAP) for which we
state two approximation algorithms. We solve the QAP in parallel to gradient
descent. This procedure acts as a surrogate loss and allows to implicitly
minimize the more expressive Earth Movers Distance (EMD) even for point clouds
with way more than $10^6$ points. As evaluating the EMD on high resolution
point clouds is intractable, we propose a divide-and-conquer approach based on
k-d trees, the EM-kD, as a scaleable and fast but still reliable upper bound
for the EMD. NeuralQAAD is demonstrated on COMA, D-FAUST, and Skulls to
significantly outperform the current state-of-the-art visually and in terms of
the EM-kD. Skulls is a novel dataset of skull CT-scans which we will make
publicly available together with our implementation of NeuralQAAD.
