  We propose VecKM, a local point cloud geometry encoder that is descriptive
and efficient to compute. VecKM leverages a unique approach by vectorizing a
kernel mixture to represent the local point cloud. Such representation's
descriptiveness is supported by two theorems that validate its ability to
reconstruct and preserve the similarity of the local shape. Unlike existing
encoders downsampling the local point cloud, VecKM constructs the local
geometry encoding using all neighboring points, producing a more descriptive
encoding. Moreover, VecKM is efficient to compute and scalable to large point
cloud inputs: VecKM reduces the memory cost from $(n^2+nKd)$ to $(nd+np)$; and
reduces the major runtime cost from computing $nK$ MLPs to $n$ MLPs, where $n$
is the size of the point cloud, $K$ is the neighborhood size, $d$ is the
encoding dimension, and $p$ is a marginal factor. The efficiency is due to
VecKM's unique factorizable property that eliminates the need of explicitly
grouping points into neighbors. In the normal estimation task, VecKM
demonstrates not only 100x faster inference speed but also highest accuracy and
strongest robustness. In classification and segmentation tasks, integrating
VecKM as a preprocessing module achieves consistently better performance than
the PointNet, PointNet++, and point transformer baselines, and runs
consistently faster by up to 10 times.
