SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1687-1734, January 2021.
Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace--Beltrami operators. Theoretical bounds on the approximation error are, however, generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localized compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace--Beltrami operator's associated PDE and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretization. We also introduce an alternative normalization for diffusion maps based on Sinkhorn weights. This normalization approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalization on flat domains, and we present a highly efficient rigorous algorithm to compute the Sinkhorn weights.

中文翻译：

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扩散图的光谱收敛：改进的误差界限和替代归一化

SIAM 数值分析杂志，第 59 卷，第 3 期，第 1687-1734 页，2021 年 1 月。
扩散图是一种广泛用于降维的流形学习算法。使用来自分布的样本，它逼近相关拉普拉斯-贝尔特拉米算子的特征值和特征函数。然而，近似误差的理论界限通常比实践中看到的速率要弱得多。本文使用新方法来改进模型案例中的误差界限，其中分布支持超圆环。对于误差的数据采样（方差）分量，我们在某些 Hardy 空间上进行空间局部紧凑嵌入估计；我们研究确定性（偏差）分量作为拉普拉斯-贝尔特拉米算子相关偏微分方程的扰动，并应用相关的光谱稳定性结果。使用这些方法，我们为频谱数据和算子离散化的范数收敛匹配长期存在的逐点误差界限。我们还介绍了基于 Sinkhorn 权重的扩散图的替代归一化。这种归一化近似于样本上的朗之万扩散，并产生对称算子近似。我们证明它与平坦域上的标准归一化相比具有更好的收敛性，并且我们提出了一种高效的严格算法来计算 Sinkhorn 权重。