# Graded persistence diagrams and persistence landscapes

## TL;DR

This paper introduces graded persistence diagrams as a refined version of persistence diagrams, providing a more discriminative and stable tool for topological data analysis, with proven bounds relating the two concepts.

## Contribution

It defines graded persistence diagrams, establishes their stability properties, and demonstrates their enhanced discriminative power over traditional persistence diagrams.

## Key findings

- The 1-Wasserstein distance bounds for graded diagrams are tighter than for persistence diagrams.
- Positive and negative points in graded diagrams correspond to local maxima and minima in persistence landscapes.
- Graded persistence diagrams are more discriminative than standard persistence diagrams.

## Abstract

We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or -1. The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the k-th graded persistence diagram correspond to the local maxima and minima, respectively, of the k-th persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between k-th graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the k-th graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams.