Given a planar curve, imagine rolling a sphere along that curve without slipping or twisting, and by this means tracing out a curve on the sphere. It is well known that such a rolling operation induces a local isometry between the sphere and the plane so that the two curves uniquely determine each other, and moreover, the operation extends to a general class of manifolds in any dimension. We use rolling to construct an analogue of a Gaussian process on a manifold starting from a Euclidean Gaussian process. The resulting model is generative, and is amenable to statistical inference given data as curves on a manifold. We illustrate with examples on the unit sphere, symmetric positive-definite matrices, and with a robotics application involving 3D orientations.

翻译：给定一条平面曲线，设想沿该曲线无滑动或无扭曲地滚动一个球体，并借此在球面上追踪出一条曲线。众所周知，这样的滚动操作会在球面与平面之间诱导出一个局部等距映射，从而使两条曲线能够唯一地相互确定；此外，该操作可推广至任意维度的一类广义流形。我们利用滚动构造了一种流形上的高斯过程类比模型，该模型始于一个欧几里得高斯过程。所得模型是生成式的，并且对于流形上以曲线形式给出的数据，适合进行统计推断。我们通过单位球面、对称正定矩阵以及涉及三维方向的机器人学应用实例加以说明。