  Persistent homology is a popular computational tool for analyzing the
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 traditional
persistent homology becomes very sensitive to noise and fails to detect the
correct topology. The same holds true for 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 to detect the correct topology even in the presence
of high-dimensional noise. Moreover, we derive a novel closed-form formula for
effective resistance, and describe its relation to diffusion distances.
Finally, we apply these methods to high-dimensional single-cell RNA-sequencing
data and show that spectral distances allow robust detection of cell cycle
loops.
