# Reconstruction of univariate functions from directional persistence diagrams

arXiv: https://arxiv.org/abs/2203.01894 · HTML: https://mlsanity.com/paper/2203.01894/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 30 references that are in the map.
- First posted on arXiv 3 Mar 2022.

## Builds on (most cited first, 6 of 6)

- [Protection Against Reconstruction and Its Applications in Private Federated Learning](https://mlsanity.com/paper/1812.00984/) (2018)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [Inverse Problems in Topological Persistence](https://mlsanity.com/paper/1810.10813/) (2018)
- [Reconstructing Embedded Graphs from Persistence Diagrams](https://mlsanity.com/paper/1912.08913/) (2019)
- [Towards Stratified Space Learning: Linearly Embedded Graphs](https://mlsanity.com/paper/2101.04375/) (2021)

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

- [Importance attribution in neural networks by means of persistence landscapes of time series](https://mlsanity.com/paper/2302.03132/) (2023)

## Abstract (arXiv)

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.