We introduce an analytic distance function for moderately sized point sets of known cardinality that is shown to have very desirable properties, both as a loss function as well as a regularizer for machine learning applications. We compare our novel construction to other point set distance functions and show proof of concept experiments for training neural networks end-to-end on point set prediction tasks such as object detection.
This paper proposes a new loss for points registration (aligning two point sets) with preferable permutation invariant property. For a 2D point set, the idea is to define a complex polynomial with the points (interpreted as complex numbers) as roots. To compare two point sets, the loss eventually boils down to evaluating the complex polynomial at all target points. The loss can be extended to high-dimensional point sets without the same theoretical framework. It is a sum over each target point the product of the distance of between the target point and all points of the source set. The claimed advantage of the proposed distance is it is more efficient compare to other metric such as Hausdorff distance or the sum of squared distance between matched point pairs (SMD). Experiments show that the proposed loss can be used in training.
The idea is interesting and new. However, the paper has a lot of room to improve. Here are a few specific issues that can be addressed in future version:
1) It is unclear what the advantage of the new idea is. I agree it leads to strong gradients compared to Hausdorff distance. However, this way of pulling all points to all points may also be very inefficient, as each point can get pulled toward all target points. These forces can easily cancel each other. 
2) As already hinted in the paper, the gradient can be extremely unstable numerically. There is not any solution to address this issue.
3) Experimental results do not show the proposed method has any advantage over others. Also important loss such as the Wasserstein distance should be considered and compared with.