  This paper explores the problem of task-oriented downsampling over 3D point
clouds, which aims to downsample a point cloud while maintaining the
performance of subsequent applications applied to the downsampled sparse points
as much as possible. Designing from the perspective of matrix optimization, we
propose MOPS-Net, a novel interpretable deep learning-based method, which is
fundamentally different from the existing deep learning-based methods due to
its interpretable feature. The optimization problem is challenging due to its
discrete and combinatorial nature. We tackle the challenges by relaxing the
binary constraint of the variables, and formulate a constrained and
differentiable matrix optimization problem. We then design a deep neural
network to mimic the matrix optimization by exploring both the local and global
structures of the input data. MOPS-Net can be end-to-end trained with a task
network and is permutation-invariant, making it robust to the input. We also
extend MOPS-Net such that a single network after one-time training is capable
of handling arbitrary downsampling ratios. Extensive experimental results show
that MOPS-Net can achieve favorable performance against state-of-the-art deep
learning-based methods over various tasks, including classification,
reconstruction, and registration. Besides, we validate the robustness of
MOPS-Net on noisy data.
