# A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis

arXiv: https://arxiv.org/abs/2109.00530 · HTML: https://mlsanity.com/paper/2109.00530/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 93 references that are in the map.
- First posted on arXiv 1 Sep 2021.

## Builds on (most cited first, 26 of 26)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [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)
- [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)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [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)
- [A mathematical model for automatic differentiation in machine learning](https://mlsanity.com/paper/2006.02080/) (2020)
- [Connectivity-Optimized Representation Learning via Persistent Homology](https://mlsanity.com/paper/1906.09003/) (2019)
- [Persistence Curves: A canonical framework for summarizing persistence diagrams](https://mlsanity.com/paper/1904.07768/) (2019)
- [Explicit topological priors for deep-learning based image segmentation using persistent homology](https://mlsanity.com/paper/1901.10244/) (2019)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Topology-Aware Surface Reconstruction for Point Clouds](https://mlsanity.com/paper/1811.12543/) (2018)
- [Statistical Analysis and Parameter Selection for Mapper](https://mlsanity.com/paper/1706.00204/) (2017)
- [On Correctness of Automatic Differentiation for Non-Differentiable Functions](https://mlsanity.com/paper/2006.06903/) (2020)
- [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)
- [Structure and Stability of the 1-Dimensional Mapper](https://mlsanity.com/paper/1511.05823/) (2015)
- [Probabilistic Convergence and Stability of Random Mapper Graphs](https://mlsanity.com/paper/1909.03488/) (2019)
- [Convergence between Categorical Representations of Reeb Space and Mapper](https://mlsanity.com/paper/1512.04108/) (2015)
- [Regularization of Persistent Homology Gradient Computation](https://mlsanity.com/paper/2011.05804/) (2020)
- [Topological Data Analysis of Single-cell Hi-C Contact Maps](https://mlsanity.com/paper/1812.01360/) (2018)
- [A numerical measure of the instability of Mapper-type algorithms](https://mlsanity.com/paper/1906.01507/) (2019)
- [Optimisation of Spectral Wavelets for Persistence-based Graph Classification](https://mlsanity.com/paper/2101.05201/) (2021)

## Built on by (most cited first, 5 of 5)

- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [Cover Learning for Large-Scale Topology Representation](https://mlsanity.com/paper/2503.09767/) (2025)
- [Cycles Communities from the Perspective of Dendrograms and Gradient Sampling](https://mlsanity.com/paper/2512.12974/) (2025)
- [Modeling high dimensional point clouds with the spherical cluster model](https://mlsanity.com/paper/2512.21960/) (2025)

## Abstract (arXiv)

We introduce a novel gradient descent algorithm extending the well-known Gradient Sampling methodology to the class of stratifiably smooth objective functions, which are defined as locally Lipschitz functions that are smooth on some regular pieces-called the strata-of the ambient Euclidean space. For this class of functions, our algorithm achieves a sub-linear convergence rate. We then apply our method to objective functions based on the (extended) persistent homology map computed over lower-star filters, which is a central tool of Topological Data Analysis. For this, we propose an efficient exploration of the corresponding stratification by using the Cayley graph of the permutation group. Finally, we provide benchmark and novel topological optimization problems, in order to demonstrate the utility and applicability of our framework.