In this paper, we extend the diffusion maps algorithm on a family of heat kernels that are either local (having exponential decay) or nonlocal (having polynomial decay), arising in various applications. For example, these kernels have been used as a regularizer in various supervised learning tasks for denoising images. Importantly, these heat kernels give rise to operators that include (but are not restricted to) the generators of the classical Laplacian associated to Brownian processes as well as the fractional Laplacian associated with β-stable Lévy processes. For local kernels, while the method is a version of the diffusion maps algorithm, we show that the applications with non-Gaussian local heat kernels approximate temporally rescaled Laplace-Beltrami operators. For the non-local heat kernels, we modify the diffusion maps algorithm to estimate fractional Laplacian operators. Here, the graph distance is used to approximate the geodesic distance with appropriate error bounds. While this approximation becomes numerically expensive as the number of data points increases, it produces an accurate operator estimation that is robust to the choice of the kernel bandwidth parameter value. In contrast, the local kernels are numerically more efficient but more sensitive to the choice of kernel bandwidth parameter value. In an application to estimate non-smooth regression functions, we find that using the nonlocal kernel as a regularizer produces a more robust and accurate estimate than using local kernels. For manifolds with boundary, we find that the proposed fractional diffusion maps framework implemented with non-local kernels approximates the regional fractional Laplacian.

在本文中，我们将扩散映射算法扩展到在各种应用中出现的局部（具有指数衰减）或非局部（具有多项式衰减）热核家族。例如，这些内核已被用作各种监督学习任务中的正则化器，以对图像进行降噪。重要的是，这些热核产生了算子，这些算子包括（但不限于）与布朗过程相关的经典拉普拉斯算子以及与β相关的分数拉普拉斯算子稳定的Lévy流程。对于局部核，虽然该方法是扩散图算法的一种版本，但我们显示具有非高斯局部热核的应用程序近似于时间上重新定标的Laplace-Beltrami算子。对于非局部热核，我们修改了扩散图算法以估计分数拉普拉斯算子。在此，图形距离用于以适当的误差范围近似测地线距离。尽管随着数据点数量的增加，这种近似在数值上变得昂贵，但它会产生对内核带宽参数值的选择具有鲁棒性的精确运算符估计。相反，局部内核在数值上更有效，但对内核带宽参数值的选择更敏感。在估算非平滑回归函数的应用中，我们发现，使用非本地内核作为正则化器比使用本地内核会产生更健壮和准确的估计。对于具有边界的流形，我们发现使用非局部核实现的拟议分数扩散图框架近似于区域分数拉普拉斯算子。