Computing persistent homology under random projection

Random projection is a tried-and-true technique in signal processing for reducing sensing complexity while maintaining acceptable performance of downstream processing tasks. In this paper, we investigate random linear projection of point clouds followed by topological data analysis for computing persistence diagrams and Betti numbers. In this first empirical study of its kind in the literature, we find that Betti numbers can be recovered accurately with high probability after random projection up to a certain reduced dimension but then the probability of recovery decreases to zero. We further investigate how the mean of the persistence diagrams from several random projections can be used favorably in Betti number recovery. Our empirical study includes both synthetic data as well as real-world range image and respiratory audio data.