Dimensionality Reduction for Persistent Homology with Gaussian Kernels

Computing persistent homology using Gaussian kernels is useful in the domains of topological data analysis and machine learning. We consider a power distance version of the Gaussian kernel distance (GKPD) given by Phillips, Wang and Zheng, and show that for high-dimensional datasets, the persistent homology of the Cech filtration computed using the GKPD is approximately preserved under dimensionality reduction using Random Fourier Features (RFFs).

In the proof, an important tool is a new concentration inequality for sums of cosines of projections of Gaussian random vectors, which we call Gaussian cosine chaoses. We believe this is of independent interest and will find other applications in future.