A universal null-distribution for topological data analysis

One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams arising from data. Despite much effort and its many successful applications, this is largely an open problem. We present a surprising discovery: normalized properly, persistence diagrams arising from random point-clouds obey a universal probability law. Our statements are based on extensive experimentation on both simulated and real data, covering point-clouds with vastly different geometry, topology, and probability distributions. Our results also include an explicit well-known distribution as a candidate for the universal law. We demonstrate the power of these new discoveries by proposing a new hypothesis testing framework for computing significance values for individual topological features within persistence diagrams, providing a new quantitative way to assess the significance of structure in data.