# Diffeomorphic interpolation for efficient persistence-based topological optimization

arXiv: https://arxiv.org/abs/2405.18820 · HTML: https://mlsanity.com/paper/2405.18820/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 37 references that are in the map.
- First posted on arXiv 29 May 2024.

## Builds on (most cited first, 22 of 22)

- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Stochastic Convergence of Persistence Landscapes and Silhouettes](https://mlsanity.com/paper/1312.0308/) (2013)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Robust Topological Inference: Distance To a Measure and Kernel Distance](https://mlsanity.com/paper/1412.7197/) (2014)
- [Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs](https://mlsanity.com/paper/2105.04404/) (2021)
- [Subsampling Methods for Persistent Homology](https://mlsanity.com/paper/1406.1901/) (2014)
- [Topological Optimization with Big Steps](https://mlsanity.com/paper/2203.16748/) (2022)
- [A Fast and Robust Method for Global Topological Functional Optimization](https://mlsanity.com/paper/2009.08496/) (2020)
- [Continuation of Point Clouds via Persistence Diagrams](https://mlsanity.com/paper/1506.03147/) (2015)
- [Adversary Detection in Neural Networks via Persistent Homology](https://mlsanity.com/paper/1711.10056/) (2017)
- [Topological Data Analysis for Arrhythmia Detection through Modular Neural Networks](https://mlsanity.com/paper/1906.05795/) (2019)
- [On the Bootstrap for Persistence Diagrams and Landscapes](https://mlsanity.com/paper/1311.0376/) (2013)
- [A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis](https://mlsanity.com/paper/2109.00530/) (2021)
- [Topological Regularization via Persistence-Sensitive Optimization](https://mlsanity.com/paper/2011.05290/) (2020)

## Built on by (most cited first, 6 of 6)

- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [Cover Learning for Large-Scale Topology Representation](https://mlsanity.com/paper/2503.09767/) (2025)
- [TopoPrune: Robust Data Pruning via Unified Latent Space Topology](https://mlsanity.com/paper/2602.02739/) (2026)
- [Barycentric Neural Networks and Length-Weighted Persistent Entropy Loss: A Green Geometric and Topological Framework for Function Approximation](https://mlsanity.com/paper/2509.06694/) (2025)
- [Persistence-Augmented Neural Networks](https://mlsanity.com/paper/2604.08469/) (2026)
- [Towards Scalable Persistence-Based Topological Optimization](https://mlsanity.com/paper/2605.10996/) (2026)

## Abstract (arXiv)

Topological Data Analysis (TDA) provides a pipeline to extract quantitative topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to what extent a given object exhibits some topological properties. These losses can then be used to perform topological optimizationvia gradient descent routines. While theoretically sounded, topological optimization faces an important challenge: gradients tend to be extremely sparse, in the sense that the loss function typically depends on only very few coordinates of the input object, yielding dramatically slow optimization schemes in practice.Focusing on the central case of topological optimization for point clouds, we propose in this work to overcome this limitation using diffeomorphic interpolation, turning sparse gradients into smooth vector fields defined on the whole space, with quantifiable Lipschitz constants. In particular, we show that our approach combines efficiently with subsampling techniques routinely used in TDA, as the diffeomorphism derived from the gradient computed on a subsample can be used to update the coordinates of the full input object, allowing us to perform topological optimization on point clouds at an unprecedented scale. Finally, we also showcase the relevance of our approach for black-box autoencoder (AE) regularization, where we aim at enforcing topological priors on the latent spaces associated to fixed, pre-trained, black-box AE models, and where we show thatlearning a diffeomorphic flow can be done once and then re-applied to new data in linear time (while vanilla topological optimization has to be re-run from scratch). Moreover, reverting the flow allows us to generate data by sampling the topologically-optimized latent space directly, yielding better interpretability of the model.