Fractional diffusion maps
Fractional diffusion maps
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分数扩散图
DOI：
10.1016/j.acha.2021.03.005
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发表时间：
2021
影响因子：
2.5
通讯作者：
Harlim, John
中科院分区：
文献类型：
--
作者：
Antil, Harbir;Berry, Tyrus;Harlim, John
在这篇文章中，我们将扩散映射算法推广到一族热核上，这些热核要么是局部的(具有指数衰减的)，要么是非局部的(具有多项式衰减的)，出现在各种应用中。例如，这些核函数已被用作图像去噪的各种监督学习任务的正则化函数。重要的是，这些热核产生的算子包括(但不限于)与布朗过程相关的经典拉普拉斯算子的生成元，以及与β稳定的Lévy过程相关的分数拉普拉斯算子。对于局部核，虽然该方法是扩散映射算法的一个版本，但我们证明了具有非高斯局部热核的应用程序近似于时间重新标度的Laplace-Beltrami算子。对于非局部热核，我们改进了扩散映射算法来估计分数次拉普拉斯算子。这里，使用图距离来近似具有适当误差界的测地线距离。虽然随着数据点数量的增加，这种近似在数值上变得昂贵，但它产生了对内核带宽参数值的选择具有健壮性的准确的算子估计。相比之下，局部核在数值上更有效，但对核带宽参数值的选择更敏感。在估计非光滑回归函数的应用中，我们发现使用非局部核作为正则化函数比使用局部核产生更稳健和更准确的估计。对于有边界的流形，我们发现所提出的非局部核实现的分数阶扩散映射框架逼近区域分数阶拉普拉斯。
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 withβ-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.