We propose a novel approach for performing convolution of signals on curved
surfaces and show its utility in a variety of geometric deep learning
applications. Key to our construction is the notion of directional functions
defined on the surface, which extend the classic real-valued signals and which
can be naturally convolved with with real-valued template functions. As a
result, rather than trying to fix a canonical orientation or only keeping the
maximal response across all alignments of a 2D template at every point of the
surface, as done in previous works, we show how information across all
rotations can be kept across different layers of the neural network. Our
construction, which we call multi-directional geodesic convolution, or
directional convolution for short, allows, in particular, to propagate and
relate directional information across layers and thus different regions on the
shape. We first define directional convolution in the continuous setting, prove
its key properties and then show how it can be implemented in practice, for
shapes represented as triangle meshes. We evaluate directional convolution in a
wide variety of learning scenarios ranging from classification of signals on
surfaces, to shape segmentation and shape matching, where we show a significant
improvement over several baselines.

本文提出一种新的方法，可以在曲面上执行信号卷积，并展示其在各种几何深度学习应用中的效用。关键在于基于曲面定义的方向性函数，可以自然地与真实值模板函数进行卷积，并在神经网络的不同层之间保留所有旋转的信息，从而允许传播和关联定向信息。

基于等变卷积的多向流动地球神经网络

Multi-directional Geodesic Neural Networks via Equivariant Convolution

This paper describes a method for efficiently computing parallel transport of
tangent vectors on curved surfaces, or more generally, any vector-valued data
on a curved manifold. More precisely, it extends a vector field defined over
any region to the rest of the domain via parallel transport along shortest
geodesics. This basic operation enables fast, robust algorithms for
extrapolating level set velocities, inverting the exponential map, computing
geometric medians and Karcher/Fr\'{e}chet means of arbitrary distributions,
constructing centroidal Voronoi diagrams, and finding consistently ordered
landmarks. Rather than evaluate parallel transport by explicitly tracing
geodesics, we show that it can be computed via a short-time heat flow involving
the connection Laplacian. As a result, transport can be achieved by solving
three prefactored linear systems, each akin to a standard Poisson problem. To
implement the method we need only a discrete connection Laplacian, which we
describe for a variety of geometric data structures (point clouds, polygon
meshes, etc). We also study the numerical behavior of our method, showing
empirically that it converges under refinement, and augment the construction of
intrinsic Delaunay triangulations (iDT) so that they can be used in the context
of tangent vector field processing.

该论文提出了一种高效计算曲面上切向量或在曲面上的任意矢量的平行运输的方法，借助于最短测地线的平行传输，通过一种基本的操作，能够快速地计算几何中值和构造中心点沃罗诺伊图等应用，并通过解决三个预分解的线性系统实现传输。