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 with mean $m$ and covariance $K$, and refer to it as a rolled Gaussian process parameterized by $m$ and $K$. The resulting model is generative, and is amenable to statistical inference given data as curves on a manifold. We identify conditions on the manifold under which the rolling of $m$ equals the Fréchet mean of the rolled Gaussian process, propose computationally simple estimators of $m$ and $K$, and derive their rates of convergence. 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，想象一个球体沿着该曲线无滑动或扭曲地滚动，并由此在球面上描绘出一条曲线。众所周知，这样的滚动操作在球面与平面之间诱导出一个局部等距映射，使得两条曲线能够唯一地相互确定，而且该操作可以推广到任意维度的一类广义流形上。我们利用滚动操作，从一个具有均值 $m$ 和协方差 $K$ 的欧几里得高斯过程出发，构造流形上的一个类比模型，并将其称为由 $m$ 和 $K$ 参数化的滚动高斯过程。所得模型是生成式的，并且适用于处理流形上曲线数据的统计推断。我们识别了流形上使得 $m$ 的滚动等于滚动高斯过程的 Fréchet 均值的条件，提出了计算简便的 $m$ 和 $K$ 估计量，并推导了它们的收敛速率。我们通过单位球面、对称正定矩阵上的例子，以及一个涉及三维方向的机器人学应用进行说明。