Reconstruction of univariate functions from directional persistence diagrams

TL;DR

This paper presents a method to reconstruct univariate functions using directional persistence diagrams, enabling efficient approximation of functions and identification of critical points with minimal directional information.

Contribution

It introduces algorithms for reconstructing functions from a small number of directional persistence diagrams, advancing the use of topological data analysis in function approximation.

Findings

Three directions suffice to locate all local extrema of piecewise linear functions.

Five directions are needed for smooth functions with non-degenerate critical points.

The method aids in reducing critical points in signals for machine learning without losing classifier information.

Abstract

We describe a method for approximating a single-variable function $f$ using persistence diagrams of sublevel sets of $f$ from height functions in different directions. We provide algorithms for the piecewise linear case and for the smooth case. Three directions suffice to locate all local maxima and minima of a piecewise linear continuous function from its collection of directional persistence diagrams, while five directions are needed in the case of smooth functions with non-degenerate critical points. Our approximation of functions by means of persistence diagrams is motivated by a study of importance attribution in machine learning, where one seeks to reduce the number of critical points of signal functions without a significant loss of information for a neural network classifier.