Projected distances for multi-parameter persistence modules

Abstract

We introduce the new notions of projected distances and projected barcodes for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto R. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). We conduct a systematic study of the stability of projected barcodes and show that the fibered barcode is a particular instance of projected barcodes. We prove that the ISM and the SCD provide lower bounds for the convolution distance. Furthermore, we show that the γ-linear ISM and the γ-linear SCD which are projected distances tailored for γ-sheaves can be computed using TDA software dedicated to one-parameter persistence modules. Moreover, the time and memory complexity required to compute these two metrics are advantageous, since our approach does not require computing nor storing an entire n-persistence module. N. B. was supported by the Swiss Innovation Agency (Innosuisse project 41665.1 IP-ICT). F. P. was supported by the

Persistence and the Sheaf-Function Correspondence

“…Also, we endow R 𝑑 with the norm • ∞ defined by 𝑥 ∞ := max 𝑖 |𝑥 𝑖 | and denote by 𝑑 𝐶 the associated convolution distance on D 𝑏 (k R 𝑑 ). Theorem 4.3 [8]. For all 𝐹, 𝐺 ∈ D 𝑏 (k R 𝑑 ), and 𝐻, 𝐼 ∈ D 𝑏 (k R 𝑑 𝛾 ) one has [7], the authors introduce a pair of adjoint functors…”

“…Inspired by persistence theory from topological data analysis (TDA) [36,21], Kashiwara and Schapira have recently introduced the convolution distance between (derived) sheaves of k-vector spaces on a finite-dimensional real normed vector space [27]. This construction has found important applications, both in TDA -where it allows expressing stability of certain constructions with respect to noise in datasets - [6,9,7,8] and in symplectic topology [2,3,23]. A challenging research direction, of interest to these two fields, is to associate numerical invariants to a sheaf on a vector space, which satisfy a certain form of continuity with respect to the convolution distance.…”