mlsanityData to 14 Oct 2026

Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds

arXiv 2402.15058 ↗First posted 23 Feb 2024math.AT cs.CG cs.LGcited by 2

Not on the map. Why?

Abstract (arXiv)

We combine standard persistent homology with image persistent homology to define a novel way of characterizing shapes and interactions between them. In particular, we introduce: (1) a mixup barcode, which captures geometric-topological interactions (mixup) between two point sets in arbitrary dimension; (2) simple summary statistics, total mixup and total percentage mixup, which quantify the complexity of the interactions as a single number; (3) a software tool for playing with the above. As a proof of concept, we apply this tool to a problem arising from machine learning. In particular, we study the disentanglement in embeddings of different classes. The results suggest that topological mixup is a useful method for characterizing interactions for low and high-dimensional data. Compared to the typical usage of persistent homology, the new tool is sensitive to the geometric locations of the topological features, which is often desirable.

Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 26 of its 57 references that are in the map. First posted on arXiv 23 Feb 2024.

Builds on

26 of 57 references.

References · 26

  1. 2021MLP-Mixer: An all-MLP Architecture for Visioncited by 1,446
  2. 2015Persistence Images: A Stable Vector Representation of Persistent Homologycited by 236
  3. 2019Topology-Preserving Deep Image Segmentationcited by 198
  4. 2019A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homologycited by 134
  5. 2019Ripser: efficient computation of Vietoris-Rips persistence barcodescited by 117
  6. 2017Deep Learning with Topological Signaturescited by 114
  7. 2020Topology of deep neural networkscited by 112
  8. 2019A Topology Layer for Machine Learningcited by 80
  9. 2018Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topologycited by 73
  10. 2021Representation Topology Divergence: A Method for Comparing Neural Network Representationscited by 66
  11. 2018A Topological Regularizer for Classifiers via Persistent Homologycited by 63
  12. 2022Topologically faithful image segmentation via induced matching of persistence barcodescited by 49
14 more
  1. 2016Geometry Helps to Compare Persistence Diagramscited by 39
  2. 2021Link Prediction with Persistent Homology: An Interactive Viewcited by 37
  3. 2021Topological Detection of Trojaned Neural Networkscited by 34
  4. 2023Topological Deep Learning: A Review of an Emerging Paradigmcited by 34
  5. 2018Exposition and Interpretation of the Topology of Neural Networkscited by 20
  6. 2020Evaluating the Disentanglement of Deep Generative Models through Manifold Topologycited by 20
  7. 2021Activation Landscapes as a Topological Summary of Neural Network Performancecited by 18
  8. 2021Topological Measurement of Deep Neural Networks Using Persistent Homologycited by 9
  9. 2021Cycle Registration in Persistent Homology with Applications in Topological Bootstrapcited by 8
  10. 2013Induced Matchings and the Algebraic Stability of Persistence Barcodescited by 7
  11. 2019TopoAct: Visually Exploring the Shape of Activations in Deep Learningcited by 4
  12. 2022Efficient Computation of Image Persistencecited by 3
  13. 2021Partial Matchings Induced by Morphisms between Persistence Modulescited by 2
  14. 2022Chromatic Alpha Complexescited by 2

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Certified Topological Interaction in Neural Representations: Exact Tests and the Statistic They Requirecited by 1
  2. 2026The Null Is the Hard Part: Exact Tests for Memorization in Generative Modelscited by 0