A derived isometry theorem for constructible sheaves on $\mathbb{R}$

TL;DR

This paper extends the isometry theorem for persistent homology into the derived sheaf-theoretic setting, enabling combinatorial and stable topological data analysis using graded barcodes.

Contribution

It proves the isometry theorem in the derived sheaf setting, relating convolution distance to graded barcodes, and explicitly computes morphisms for distance calculations.

Findings

Convolution distance is shown to be closed.

Connected components of the derived category are characterized.

Explicit examples of distance computations are provided.

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 link between persistent homology and sheaves. We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To achieve so, we explicitly compute all morphisms in $D^b_{\mathbb{R} c}(\textbf{k}_\mathbb{R})$, which enables us to compute distances between indecomposable objects. Then we adapt Bjerkevik's stability proof to this derived setting. As a byproduct of our isometry theorem, we prove that the convolution distance is closed, give a precise description of connected components of $D^b_{\mathbb{R} c}(\textbf{k}_\mathbb{R})$and provide some explicit examples of computation of the convolution distance.