# Manifold Topology Divergence: a Framework for Comparing Data Manifolds

arXiv: https://arxiv.org/abs/2106.04024 · HTML: https://mlsanity.com/paper/2106.04024/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 45 references that are in the map.
- First posted on arXiv 8 Jun 2021.

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

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Conditional Image Synthesis With Auxiliary Classifier GANs](https://mlsanity.com/paper/1610.09585/) (2016)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [Are GANs Created Equal? A Large-Scale Study](https://mlsanity.com/paper/1711.10337/) (2017)
- [Pros and Cons of GAN Evaluation Measures](https://mlsanity.com/paper/1802.03446/) (2018)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Classification Accuracy Score for Conditional Generative Models](https://mlsanity.com/paper/1905.10887/) (2019)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [CovidGAN: Data Augmentation Using Auxiliary Classifier GAN for Improved Covid-19 Detection](https://mlsanity.com/paper/2103.05094/) (2021)
- [Geometry Score: A Method For Comparing Generative Adversarial Networks](https://mlsanity.com/paper/1802.02664/) (2018)
- [The Shape of Data: Intrinsic Distance for Data Distributions](https://mlsanity.com/paper/1905.11141/) (2019)
- [GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes](https://mlsanity.com/paper/2003.07989/) (2020)
- [Skill Rating for Generative Models](https://mlsanity.com/paper/1808.04888/) (2018)

## Built on by (most cited first, 30 of 31)

- [Similarity of Neural Network Models: A Survey of Functional and Representational Measures](https://mlsanity.com/paper/2305.06329/) (2023)
- [Intrinsic Dimension Estimation for Robust Detection of AI-Generated Texts](https://mlsanity.com/paper/2306.04723/) (2023)
- [Unleashing Transformers: Parallel Token Prediction with Discrete Absorbing Diffusion for Fast High-Resolution Image Generation from Vector-Quantized Codes](https://mlsanity.com/paper/2111.12701/) (2021)
- [Learning Topology-Preserving Data Representations](https://mlsanity.com/paper/2302.00136/) (2023)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [Hallucination Detection in LLMs with Topological Divergence on Attention Graphs](https://mlsanity.com/paper/2504.10063/) (2025)
- [TopP&R: Robust Support Estimation Approach for Evaluating Fidelity and Diversity in Generative Models](https://mlsanity.com/paper/2306.08013/) (2023)
- [A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Science](https://mlsanity.com/paper/2403.12636/) (2024)
- [Position: All Current Generative Fidelity and Diversity Metrics are Flawed](https://mlsanity.com/paper/2505.22450/) (2025)
- [Uncertainty Estimation of Transformers' Predictions via Topological Analysis of the Attention Matrices](https://mlsanity.com/paper/2308.11295/) (2023)
- [Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets](https://mlsanity.com/paper/2202.01671/) (2022)
- [Mapping the Multiverse of Latent Representations](https://mlsanity.com/paper/2402.01514/) (2024)
- [Deep Regression Representation Learning with Topology](https://mlsanity.com/paper/2404.13904/) (2024)
- [Bridging the Dimensional Chasm: Uncover Layer-wise Dimensional Reduction in Transformers through Token Correlation](https://mlsanity.com/paper/2503.22547/) (2025)
- [Offline RL with Smooth OOD Generalization in Convex Hull and its Neighborhood](https://mlsanity.com/paper/2506.08417/) (2025)
- [Manifold Characteristics That Predict Downstream Task Performance](https://mlsanity.com/paper/2205.07477/) (2022)
- [Defining Neural Network Architecture through Polytope Structures of Dataset](https://mlsanity.com/paper/2402.02407/) (2024)
- [Topological Parallax: A Geometric Specification for Deep Perception Models](https://mlsanity.com/paper/2306.11835/) (2023)
- [A survey of manifold learning and its applications for multimedia](https://mlsanity.com/paper/2310.12986/) (2023)
- [Towards Scalable Topological Regularizers](https://mlsanity.com/paper/2501.14641/) (2025)
- [Topology-Aware CLIP Few-Shot Learning](https://mlsanity.com/paper/2505.01694/) (2025)
- [STEB: In Search of the Best Evaluation Approach for Synthetic Time Series](https://mlsanity.com/paper/2505.21160/) (2025)
- [The Density of Cross-Persistence Diagrams and Its Applications](https://mlsanity.com/paper/2603.11623/) (2026)
- [Scalar Function Topology Divergence: Comparing Topology of 3D Objects](https://mlsanity.com/paper/2407.08364/) (2024)
- [MAYA: Addressing Inconsistencies in Generative Password Guessing through a Unified Benchmark](https://mlsanity.com/paper/2504.16651/) (2025)
- [CoreSPECT: Enhancing Clustering Algorithms via an Interplay of Density and Geometry](https://mlsanity.com/paper/2507.08243/) (2025)
- [Generating Graph-Like Logical Rules for Knowledge Graph Reasoning via Diffusion Models](https://mlsanity.com/paper/2605.30747/) (2026)
- [Symmetric Divergence and Normalized Similarity: A Unified Topological Framework for Representation Analysis](https://mlsanity.com/paper/2606.06342/) (2026)
- [ASK-NN: An Asymmetric Nearest-Neighbor Test that detects Distribution Drifts in Natural Language](https://mlsanity.com/paper/2607.15607/) (2026)
- [CodeTD: Topology of Attention Detects Hallucinations in Code LLMs](https://mlsanity.com/paper/2609.07779/) (2026)

## Abstract (arXiv)

We develop a framework for comparing data manifolds, aimed, in particular, towards the evaluation of deep generative models. We describe a novel tool, Cross-Barcode(P,Q), that, given a pair of distributions in a high-dimensional space, tracks multiscale topology spacial discrepancies between manifolds on which the distributions are concentrated. Based on the Cross-Barcode, we introduce the Manifold Topology Divergence score (MTop-Divergence) and apply it to assess the performance of deep generative models in various domains: images, 3D-shapes, time-series, and on different datasets: MNIST, Fashion MNIST, SVHN, CIFAR10, FFHQ, chest X-ray images, market stock data, ShapeNet. We demonstrate that the MTop-Divergence accurately detects various degrees of mode-dropping, intra-mode collapse, mode invention, and image disturbance. Our algorithm scales well (essentially linearly) with the increase of the dimension of the ambient high-dimensional space. It is one of the first TDA-based practical methodologies that can be applied universally to datasets of different sizes and dimensions, including the ones on which the most recent GANs in the visual domain are trained. The proposed method is domain agnostic and does not rely on pre-trained networks.