# Towards Scalable Topological Regularizers

arXiv: https://arxiv.org/abs/2501.14641 · HTML: https://mlsanity.com/paper/2501.14641/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 89 references that are in the map.
- First posted on arXiv 24 Jan 2025.

## Builds on (most cited first, 30 of 57)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Deep CORAL: Correlation Alignment for Deep Domain Adaptation](https://mlsanity.com/paper/1607.01719/) (2016)
- [Deep Transfer Learning with Joint Adaptation Networks](https://mlsanity.com/paper/1605.06636/) (2016)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Return of Frustratingly Easy Domain Adaptation](https://mlsanity.com/paper/1511.05547/) (2015)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Rethinking FID: Towards a Better Evaluation Metric for Image Generation](https://mlsanity.com/paper/2401.09603/) (2023)
- [Learning to Generate Novel Domains for Domain Generalization](https://mlsanity.com/paper/2007.03304/) (2020)
- [clDice -- A Novel Topology-Preserving Loss Function for Tubular Structure Segmentation](https://mlsanity.com/paper/2003.07311/) (2020)
- [A Survey on Deep Semi-supervised Learning](https://mlsanity.com/paper/2103.00550/) (2021)
- [Generalized Zero- and Few-Shot Learning via Aligned Variational Autoencoders](https://mlsanity.com/paper/1812.01784/) (2018)
- [Exposing flaws of generative model evaluation metrics and their unfair treatment of diffusion models](https://mlsanity.com/paper/2306.04675/) (2023)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Learning to Diversify for Single Domain Generalization](https://mlsanity.com/paper/2108.11726/) (2021)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [The Cramer Distance as a Solution to Biased Wasserstein Gradients](https://mlsanity.com/paper/1705.10743/) (2017)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Interaction Matters: A Note on Non-asymptotic Local Convergence of Generative Adversarial Networks](https://mlsanity.com/paper/1802.06132/) (2018)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Geometry Score: A Method For Comparing Generative Adversarial Networks](https://mlsanity.com/paper/1802.02664/) (2018)

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

- [Topology-Aware Representation Alignment for Semi-Supervised Vision-Language Learning](https://mlsanity.com/paper/2604.26370/) (2026)

## Abstract (arXiv)

Latent space matching, which consists of matching distributions of features in latent space, is a crucial component for tasks such as adversarial attacks and defenses, domain adaptation, and generative modelling. Metrics for probability measures, such as Wasserstein and maximum mean discrepancy, are commonly used to quantify the differences between such distributions. However, these are often costly to compute, or do not appropriately take the geometric and topological features of the distributions into consideration. Persistent homology is a tool from topological data analysis which quantifies the multi-scale topological structure of point clouds, and has recently been used as a topological regularizer in learning tasks. However, computation costs preclude larger scale computations, and discontinuities in the gradient lead to unstable training behavior such as in adversarial tasks. We propose the use of principal persistence measures, based on computing the persistent homology of a large number of small subsamples, as a topological regularizer. We provide a parallelized GPU implementation of this regularizer, and prove that gradients are continuous for smooth densities. Furthermore, we demonstrate the efficacy of this regularizer on shape matching, image generation, and semi-supervised learning tasks, opening the door towards a scalable regularizer for topological features.