Title: Sheaves as computable and stable topological invariants for datasets: From level-sets persistence and beyond
Abstract: Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira after J. Curry has made the first links between persistent homology and sheaves. The theory of constructible sheaves over a normed vector space admits a lot of similarities with the theory of persistence modules. In this (derived) setting, Kashiwara-Schapira have introduced the convolution distance, inspired by the interleaving distance introduced. The convolution distance satisfies a stability property with respect to the L∞-norm in the same fashion as the interleaving distance. Moreover if the normed vector space is one dimensional, constructible sheaves admit a graded-barcode (like barcodes for one-parameter persistence modules).
After having quickly introduced this setting and these results, we will make explicit the connections between level-sets persistence and derived sheaf theory over the real line. In particular we construct a functor from 2-parameter persistence modules to sheaves over R, as well as a functor in the other direction. We also observe that the 2-parameter persistence modules arising from the level sets of continuous functions carry extra structure that we call a Mayer-Vietoris system. We prove classification, barcode decomposition, and stability theorems for these Mayer-Vietoris systems, and we show that the aforementioned functors establish a pseudo-isometric equivalence of categories between derived constructible sheaves with the convolution or (derived) bottleneck distance and the interleaving distance of strictly pointwise finite-dimensional Mayer-Vietoris systems. Ultimately, our results provide a functorial equivalence between level-sets persistence and derived pushforward of sheaves for continuous real-valued functions. Altogether, these results show that we can use constructible sheaves as a continuous generalization of level-sets persistence in a computer-friendly way.
