Persistence Images: A Stable Vector Representation of Persistent Homology. Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.

References in zbMATH (referenced in 40 articles , 1 standard article )

Showing results 1 to 20 of 40.
Sorted by year (citations)

1 2 next

  1. Beshkov, Kosio; Tiesinga, Paul: Geodesic-based distance reveals nonlinear topological features in neural activity from mouse visual cortex (2022)
  2. Chung, Yu-Min; Lawson, Austin: Persistence curves: a canonical framework for summarizing persistence diagrams (2022)
  3. Grbić, Jelena; Wu, Jie; Xia, Kelin; Wei, Guo-Wei: Aspects of topological approaches for data science (2022)
  4. McCleary, Alexander; Patel, Amit: Edit distance and persistence diagrams over lattices (2022)
  5. Oballe, Christopher; Cherne, Alan; Boothe, Dave; Kerick, Scott; Franaszczuk, Piotr J.; Maroulas, Vasileios: Bayesian topological signal processing (2022)
  6. Xian, Lu; Adams, Henry; Topaz, Chad M.; Ziegelmeier, Lori: Capturing dynamics of time-varying data via topology (2022)
  7. Calcina, Sabrina S.; Gameiro, Marcio: Parameter estimation in systems exhibiting spatially complex solutions via persistent homology and machine learning (2021)
  8. Chazal, Frédéric; Levrard, Clément; Royer, Martin: Clustering of measures via mean measure quantization (2021)
  9. Ciocanel, Maria-Veronica; Juenemann, Riley; Dawes, Adriana T.; McKinley, Scott A.: Topological data analysis approaches to uncovering the timing of ring structure onset in filamentous networks (2021)
  10. Divol, Vincent; Lacombe, Théo: Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport (2021)
  11. Dong, Zhetong; Pu, Junyu; Lin, Hongwei: Multiscale persistent topological descriptor for porous structure retrieval (2021)
  12. Feng, Michelle; Porter, Mason A.: Persistent homology of geospatial data: a case study with voting (2021)
  13. Padellini, Tullia; Brutti, Pierpaolo: Supervised learning with indefinite topological kernels (2021)
  14. Tymochko, Sarah; Singhal, Kritika; Heo, Giseon: Classifying sleep states using persistent homology and Markov chains: a pilot study (2021)
  15. Adams, Henry; Aminian, Manuchehr; Farnell, Elin; Kirby, Michael; Mirth, Joshua; Neville, Rachel; Peterson, Chris; Shonkwiler, Clayton: A fractal dimension for measures via persistent homology (2020)
  16. Bendich, Paul; Bubenik, Peter; Wagner, Alexander: Stabilizing the unstable output of persistent homology computations (2020)
  17. Berry, Eric; Chen, Yen-Chi; Cisewski-Kehe, Jessi; Fasy, Brittany Terese: Functional summaries of persistence diagrams (2020)
  18. Bubenik, Peter: The persistence landscape and some of its properties (2020)
  19. Cang, Zixuan; Munch, Elizabeth; Wei, Guo-Wei: Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis (2020)
  20. Cang, Zixuan; Wei, Guo-Wei: Persistent cohomology for data with multicomponent heterogeneous information (2020)

1 2 next