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.