Reconstruction of univariate functions from directional persistence diagrams

2022/03/03 by Aina Ferrà, Ferrà, Aina, Carles Casacuberta +3
Computer Science · #55N31 #62R40 #68T07 #Algebraic Topology (math.AT) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2203.01894

openalex publication_date 2022/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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