Persistent homology is a popular computational tool for detecting non-trivial
topology of point clouds, such as the presence of loops or voids. However, many
real-world datasets with low intrinsic dimensionality reside in an ambient
space of much higher dimensionality. We show that in this case vanilla
persistent homology becomes very sensitive to noise and fails to detect the
correct topology. The same holds true for most existing refinements of
persistent homology. As a remedy, we find that spectral distances on the
$k$-nearest-neighbor graph of the data, such as diffusion distance and
effective resistance, allow persistent homology to detect the correct topology
even in the presence of high-dimensional noise. Furthermore, we derive a novel
closed-form expression for effective resistance in terms of the
eigendecomposition of the graph Laplacian, and describe its relation to
diffusion distances. Finally, we apply these methods to several
high-dimensional single-cell RNA-sequencing datasets and show that spectral
distances on the $k$-nearest-neighbor graph allow robust detection of cell
cycle loops.

在高维噪声的存在下，通过 $k$ 最近邻图上的谱距离，如扩散距离和有效电阻，可以使持久同调在检测正确拓扑结构方面具有鲁棒性。该研究还导出了有效电阻的新的闭式表达式，并描述了其与扩散距离的关系。最后，通过应用这些方法于几个高维单细胞 RNA 测序数据集，表明 $k$ 最近邻图上的谱距离可以稳健地检测细胞周期环。