Modeling and replicating statistical topology and evidence for CMB nonhomogeneity

Robert J. Adler*, Sarit Agami, Pratyush Pranav

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

Under the banner of “big data,” the detection and classification of structure in extremely large, high-dimensional, data sets are two of the central statistical challenges of our times. Among the most intriguing new approaches to this challenge is “TDA,” or “topological data analysis,” one of the primary aims of which is providing nonmetric, but topologically informative, preanalyses of data which make later, more quantitative, analyses feasible. While TDA rests on strong mathematical foundations from topology, in applications, it has faced challenges due to difficulties in handling issues of statistical reliability and robustness, often leading to an inability to make scientific claims with verifiable levels of statistical confidence. We propose a methodology for the parametric representation, estimation, and replication of persistence diagrams, the main diagnostic tool of TDA. The power of the methodology lies in the fact that even if only one persistence diagram is available for analysis—the typical case for big data applications—the replications permit conventional statistical hypothesis testing. The methodology is conceptually simple and computationally practical, and provides a broadly effective statistical framework for persistence diagram TDA analysis. We demonstrate the basic ideas on a toy example, and the power of the parametric approach to TDA modeling in an analysis of cosmic microwave background (CMB) nonhomogeneity.

Original languageAmerican English
Pages (from-to)11878-11883
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Volume114
Issue number45
DOIs
StatePublished - 7 Nov 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017, National Academy of Sciences. All rights reserved.

Keywords

  • CMB nonhomogeneity
  • Gibbs measures
  • Persistence diagrams
  • Statistical topology
  • Topological data analysis

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