We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry \& Harlim, \emph{Appl.\ Comput.\ Harmon.\ Anal.}, 40(1):68--96, 2016] computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be applied in high dimensions or on geometrically complex manifolds when spatial discretizations are not available. We also employ an efficient $k$-$d$ tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.

翻译：我们采用以内核为基础的方法,从概率分布样本到接近一个多管的科尔莫戈罗夫操作员。自调变量带宽内核法[Berry {Harlim,\ emph{Appl\Comput.\ Harmon.\ Harmon.\ Anal.},40(1):68-96, 2016]计算出一个庞大的、稀疏的、接近差分操作员的矩阵。在这里,我们使用离散的eigendecomposition到 (i) 将操作员倒转,解决一个差异方程式,和(ii) 代表多管的梯度矢量字段。这些方法只要求从底部分布中提取样本,因此,在空间离散无可用的情况下,可以在高尺寸或几何学复杂式多管上应用。我们还使用高效的$-k$-d$的树算法来计算稀薄内核矩阵,这是一个计算瓶颈。