Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-29T21:44:37.912Z Has data issue: false hasContentIssue false
  • English
  • Français

Learning from the Shape of Data

Published online by Cambridge University Press:  01 January 2022

Sarita Rosenstock
Get access Rights & Permissions [Opens in a new window]

Abstract

To make sense of large data sets, we often look for patterns in how data points are “shaped” in the space of possible measurement outcomes. The emerging field of topological data analysis (TDA) offers a toolkit for formalizing the process of identifying such shapes. This article aims to discover why and how the resulting analysis should be understood as reflecting significant features of the systems that generated the data. I argue that a particular feature of TDA—its functoriality—is what enables TDA to translate visual intuitions about structure in data into precise, computationally tractable descriptions of real-world systems.

Type
Computer Simulation and Computer Science
Information
Philosophy of Science , Volume 88 , Issue 5 , December 2021 , pp. 1033 - 1044
Copyright
Copyright 2021 by the Philosophy of Science Association. All rights reserved.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bendich, P., Marron, J. S., Miller, E., Pieloch, A., and Skwerer, S.. 2016. “Persistent Homology Analysis of Brain Artery Trees.” Annals of Applied Statistics 10 (1): 198–218.CrossRefGoogle ScholarPubMed
Boyd, R. 1999. “Homeostasis, Species, and Higher Taxa.” In Species: New Inter-disciplinary Essays, ed. Wilson, R. A., 141–85. Cambridge, MA: MIT Press.Google Scholar
Bubenik, P. 2015. “Statistical Topological Data Analysis Using Persistence Landscapes.” Journal of Machine Learning Research 16:77–102.Google Scholar
Bubenik, P., and Scott, J. A.. 2014. “Categorification of Persistent Homology.” Discrete and Computational Geometry 51 (3): 600–627.CrossRefGoogle Scholar
Carlsson, G. 2009. “Topology and Data.” Bulletin of the American Mathematical Society 46 (2): 255–308.CrossRefGoogle Scholar
Carlsson, G., and Mémoli, F.. 2013. “Classifying Clustering Schemes.” Foundations of Computational Mathematics 13 (2): 221–52.CrossRefGoogle Scholar
Chazal, F., Fasy, B. T., Lecci, F., Rinaldo, A., Singh, A., and Wasserman, L.. 2015. “On the Bootstrap for Persistence Diagrams and Landscapes.” Modeling and Analysis of Information Systems 20 (6): 111–20.CrossRefGoogle Scholar
Cohen-Steiner, D., Edelsbrunner, H., and Harer, J.. 2007. “Stability of Persistence Diagrams.” Discrete and Computational Geometry 37 (1): 103–20.CrossRefGoogle Scholar
Edelsbrunner, H., Letscher, D., and Zomorodian, A.. 2002. “Topological Persistence and Simplification.” Discrete and Computational Geometry 28:511–33.CrossRefGoogle Scholar
Ghrist, R. 2008. “Barcodes: The Persistent Topology of Data.” Bulletin of the American Mathematical Society 45 (1): 61–75.CrossRefGoogle Scholar
Halvorson, H. 2013. “The Semantic View, If Plausible, Is Syntactic.” Philosophy of Science 80 (3): 475–78.CrossRefGoogle Scholar
Hatcher, A. 2002. Algebraic Topology. Cambridge: Cambridge University Press.Google Scholar
Manders, K. 2008. “The Euclidean Diagram, 1995.” In The Philosophy of Mathematical Practice. Oxford: Oxford University Press.Google Scholar
Mumma, J. 2010. “Proofs, Pictures, and Euclid.” Synthese 175 (2): 255–87.CrossRefGoogle Scholar
Nicolau, M., Levine, A. J., and Carlsson, G.. 2011. “Topology Based Data Analysis Identifies a Subgroup of Breast Cancers with a Unique Mutational Profile and Excellent Survival.” Proceedings of the National Academy of Sciences 108 (17): 7265–70.10.1073/pnas.1102826108CrossRefGoogle ScholarPubMed
Perea, J. A., and Harer, J.. 2015. “Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis.” Foundations of Computational Mathematics 15 (3): 799–838.CrossRefGoogle Scholar
Rosenstock, S. 2019. “A Categorical Consideration of Physical Formalisms.” PhD diss., University of California, Irvine.Google Scholar
Rosenstock, S.. 2021. “Clustering Schemes for Diverse Data Models.” Unpublished manuscript, Australian National University.Google Scholar
van de Weygaert, R., et al. 2011. “Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web.” In Transactions on Computational Science XIV: Special Issue on Voronoi Diagrams and Delaunay Triangulation, ed. Gavrilova, M. L., Tan, C. J. K., and Mostafavi, M. A., 60–101. Berlin: Springer.CrossRefGoogle Scholar
Weatherall, J. O. 2017. “Categories and the Foundations of Classical Field Theories.” In Categories for the Working Philosopher, ed. Landry, E., 329–48. Oxford: Oxford University Press.Google Scholar