# A Practical Solver for Scalar Data Topological Simplification

arXiv: https://arxiv.org/abs/2407.12399 · HTML: https://mlsanity.com/paper/2407.12399/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 94 references that are in the map.
- First posted on arXiv 17 Jul 2024.

## Builds on (most cited first, 20 of 20)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Topology-Aware Surface Reconstruction for Point Clouds](https://mlsanity.com/paper/1811.12543/) (2018)
- [Topological Optimization with Big Steps](https://mlsanity.com/paper/2203.16748/) (2022)
- [Task-based Augmented Contour Trees with Fibonacci Heaps](https://mlsanity.com/paper/1902.04805/) (2019)
- [Progressive Wasserstein Barycenters of Persistence Diagrams](https://mlsanity.com/paper/1907.04565/) (2019)
- [A Fast and Robust Method for Global Topological Functional Optimization](https://mlsanity.com/paper/2009.08496/) (2020)
- [Ranking Viscous Finger Simulations to an Acquired Ground Truth with Topology-aware Matchings](https://mlsanity.com/paper/1908.07841/) (2019)
- [Distributed computation of persistent homology](https://mlsanity.com/paper/1310.0710/) (2013)
- [Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data -- An Algorithm and A Benchmark](https://mlsanity.com/paper/2206.13932/) (2022)
- [Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study](https://mlsanity.com/paper/2207.14080/) (2022)
- [Localized Topological Simplification of Scalar Data](https://mlsanity.com/paper/2009.00083/) (2020)
- [Topological data analysis of vortices in the magnetically-induced current density in LiH molecule](https://mlsanity.com/paper/2212.08690/) (2022)
- [Parallel Computation of Piecewise Linear Morse-Smale Segmentations](https://mlsanity.com/paper/2303.15491/) (2023)
- [Topological Regularization via Persistence-Sensitive Optimization](https://mlsanity.com/paper/2011.05290/) (2020)
- [Accelerating Iterated Persistent Homology Computations with Warm Starts](https://mlsanity.com/paper/2108.05022/) (2021)
- [Geometry-Aware Planar Embedding of Treelike Structures](https://mlsanity.com/paper/2202.10551/) (2022)

## Built on by (most cited first, 2 of 2)

- [Topology Aware Neural Interpolation of Scalar Fields](https://mlsanity.com/paper/2508.17995/) (2025)
- [Operational Abstractions of Neural Network Concepts via Topological Representations](https://mlsanity.com/paper/2610.05017/) (2026)

## Abstract (arXiv)

This paper presents a practical approach for the optimization of topological simplification, a central pre-processing step for the analysis and visualization of scalar data. Given an input scalar field f and a set of "signal" persistence pairs to maintain, our approach produces an output field g that is close to f and which optimizes (i) the cancellation of "non-signal" pairs, while (ii) preserving the "signal" pairs. In contrast to pre-existing simplification algorithms, our approach is not restricted to persistence pairs involving extrema and can thus address a larger class of topological features, in particular saddle pairs in three-dimensional scalar data. Our approach leverages recent generic persistence optimization frameworks and extends them with tailored accelerations specific to the problem of topological simplification. Extensive experiments report substantial accelerations over these frameworks, thereby making topological simplification optimization practical for real-life datasets. Our approach enables a direct visualization and analysis of the topologically simplified data, e.g., via isosurfaces of simplified topology (fewer components and handles). We apply our approach to the extraction of prominent filament structures in three-dimensional data. Specifically, we show that our pre-simplification of the data leads to practical improvements over standard topological techniques for removing filament loops. We also show how our approach can be used to repair genus defects in surface processing. Finally, we provide a C++ implementation for reproducibility purposes.