mlsanityData to 7 Oct 2026

Manifold Topology Divergence: a Framework for Comparing Data Manifolds

arXiv 2106.04024 ↗First posted 8 Jun 2021cs.LG cs.AI cs.CVcited by 31

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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.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). Builds on 20 of its 45 references that are in the map. First posted on arXiv 8 Jun 2021.

Builds on

20 of 45 references.

References · 20

  1. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 5,986
  2. 2019Analyzing and Improving the Image Quality of StyleGANcited by 3,580
  3. 2016Improved Techniques for Training GANscited by 2,938
  4. 2015ShapeNet: An Information-Rich 3D Model Repositorycited by 2,880
  5. 2018Demystifying MMD GANscited by 1,268
  6. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  7. 2019Improved Precision and Recall Metric for Assessing Generative Modelscited by 925
  8. 2016Conditional Image Synthesis With Auxiliary Classifier GANscited by 562
  9. 2015A note on the evaluation of generative modelscited by 291
  10. 2017Are GANs Created Equal? A Large-Scale Studycited by 267
  11. 2018Pros and Cons of GAN Evaluation Measurescited by 224
  12. 2017An introduction to Topological Data Analysis: fundamental and practical aspects for data scientistscited by 171
8 more
  1. 2019Classification Accuracy Score for Conditional Generative Modelscited by 134
  2. 2019Topological Autoencoderscited by 124
  3. 2019Ripser: efficient computation of Vietoris-Rips persistence barcodescited by 115
  4. 2021CovidGAN: Data Augmentation Using Auxiliary Classifier GAN for Improved Covid-19 Detectioncited by 64
  5. 2018Geometry Score: A Method For Comparing Generative Adversarial Networkscited by 56
  6. 2019The Shape of Data: Intrinsic Distance for Data Distributionscited by 38
  7. 2020GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodescited by 26
  8. 2018Skill Rating for Generative Modelscited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 3; 2023: 7; 2024: 5; 2025: 9; 2026: 6.

31 papers.

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  1. 2023Similarity of Neural Network Models: A Survey of Functional and Representational Measurescited by 128
  2. 2023Intrinsic Dimension Estimation for Robust Detection of AI-Generated Textscited by 111
  3. 2021Unleashing Transformers: Parallel Token Prediction with Discrete Absorbing Diffusion for Fast High-Resolution Image Generation from Vector-Quantized Codescited by 71
  4. 2023Learning Topology-Preserving Data Representationscited by 30
  5. 2022Wasserstein Iterative Networks for Barycenter Estimationcited by 23
  6. 2025Hallucination Detection in LLMs with Topological Divergence on Attention Graphscited by 18
  7. 2024A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Sciencecited by 14
  8. 2023TopP&R: Robust Support Estimation Approach for Evaluating Fidelity and Diversity in Generative Modelscited by 13
  9. 2025Position: All Current Generative Fidelity and Diversity Metrics are Flawedcited by 12
  10. 2023Uncertainty Estimation of Transformers' Predictions via Topological Analysis of the Attention Matricescited by 8
  11. 2022Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasetscited by 5
  12. 2024Mapping the Multiverse of Latent Representationscited by 5
19 more
  1. 2024Deep Regression Representation Learning with Topologycited by 4
  2. 2025Bridging the Dimensional Chasm: Uncover Layer-wise Dimensional Reduction in Transformers through Token Correlationcited by 3
  3. 2025Offline RL with Smooth OOD Generalization in Convex Hull and its Neighborhoodcited by 3
  4. 2022Manifold Characteristics That Predict Downstream Task Performancecited by 2
  5. 2024Defining Neural Network Architecture through Polytope Structures of Datasetcited by 2
  6. 2023Topological Parallax: A Geometric Specification for Deep Perception Modelscited by 1
  7. 2023A survey of manifold learning and its applications for multimediacited by 1
  8. 2025Towards Scalable Topological Regularizerscited by 1
  9. 2025Topology-Aware CLIP Few-Shot Learningcited by 1
  10. 2026The Density of Cross-Persistence Diagrams and Its Applicationscited by 1
  11. 2024Scalar Function Topology Divergence: Comparing Topology of 3D Objectscited by 0
  12. 2025MAYA: Addressing Inconsistencies in Generative Password Guessing through a Unified Benchmarkcited by 0
  13. 2025STEB: In Search of the Best Evaluation Approach for Synthetic Time Seriescited by 0
  14. 2025CoreSPECT: Enhancing Clustering Algorithms via an Interplay of Density and Geometrycited by 0
  15. 2026Generating Graph-Like Logical Rules for Knowledge Graph Reasoning via Diffusion Modelscited by 0
  16. 2026Symmetric Divergence and Normalized Similarity: A Unified Topological Framework for Representation Analysiscited by 0
  17. 2026ASK-NN: An Asymmetric Nearest-Neighbor Test that detects Distribution Drifts in Natural Languagecited by 0
  18. 2026CodeTD: Topology of Attention Detects Hallucinations in Code LLMscited by 0
  19. 2026The Null Is the Hard Part: Exact Tests for Memorization in Generative Modelscited by 0