# Projected distances for multi-parameter persistence modules

arXiv: https://arxiv.org/abs/2206.08818 · HTML: https://mlsanity.com/paper/2206.08818/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2022.

## Built on by (most cited first, 1 of 1)

- [Singular Value Representation: A New Graph Perspective On Neural Networks](https://mlsanity.com/paper/2302.08183/) (2023)

## Abstract (arXiv)

Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances 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.