扩散映射的谱收敛：改进的误差界和另一种归一化方法
Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation
摘要：
扩散映射是一种广泛用于降维的流形学习算法。使用一个样本的分布，它近似的本征值和本征函数相关的拉普拉斯-贝尔特拉米运营商。然而，近似误差的理论界限通常比实践中看到的速率弱得多。本文使用新的方法来提高误差界的模型的情况下，分布是支持的超环面。对于数据采样（方差）组件的错误，我们空间局部化的紧凑嵌入估计某些哈代空间，我们研究的确定性（偏差）组件作为一个扰动的拉普拉斯-贝尔特拉米算子的相关PDE，并适用于相关的谱稳定性结果。使用这些方法，我们匹配长期存在的逐点误差界的光谱数据和规范收敛的操作员离散化。我们还介绍了另一种基于Sinkhorn权重的扩散图归一化。这种归一化近似于样品上的朗之万扩散，并产生对称算子近似。我们证明了它具有更好的收敛性相比，在平坦域上的标准规范化，并提出了一个高效的算法来计算Sinkhorn权重。
英文摘要：
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 localised 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 discretisation. We also introduce an alternative normalisation for diffusion maps based on Sinkhorn weights. This normalisation approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalisation on flat domains, and present a highly efficient algorithm to compute the Sinkhorn weights.
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