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  Persistence barcodes versus Kolmogorov signatures: Detecting modes of one-dimensional signals.

Bauer, U., Munk, A., Sieling, H., & Wardetzky, M. (2017). Persistence barcodes versus Kolmogorov signatures: Detecting modes of one-dimensional signals. Foundations of Computational Mathematics, 17(1), 1-33. doi:10.1007/s10208-015-9281-9.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-002C-DE41-7 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-002C-DE44-1
Genre: Journal Article

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2416948.pdf (Publisher version), 2MB
 
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 Creators:
Bauer, U., Author
Munk, A.1, Author              
Sieling, H., Author
Wardetzky, M., Author
Affiliations:
1Research Group of Statistical Inverse-Problems in Biophysics, MPI for biophysical chemistry, Max Planck Society, ou_1113580              

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Free keywords: Persistent homology; Mode hunting; Exponential deviation bound; Partial sum process; Taut strings
 Abstract: We investigate the problem of estimating the number of modes (i.e., local maxima)—a well-known question in statistical inference—and we show how to do so without presmoothing the data. To this end, we modify the ideas of persistence barcodes by first relating persistence values in dimension one to distances (with respect to the supremum norm) to the sets of functions with a given number of modes, and subsequently working with norms different from the supremum norm. As a particular case, we investigate the Kolmogorov norm. We argue that this modification has certain statistical advantages. We offer confidence bands for the attendant Kolmogorov signatures, thereby allowing for the selection of relevant signatures with a statistically controllable error. As a result of independent interest, we show that taut strings minimize the number of critical points for a very general class of functions. We illustrate our results by several numerical examples.

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Language(s): eng - English
 Dates: 2015-08-192017-02
 Publication Status: Published in print
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 Rev. Method: Peer
 Identifiers: DOI: 10.1007/s10208-015-9281-9
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Title: Foundations of Computational Mathematics
Source Genre: Journal
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Pages: - Volume / Issue: 17 (1) Sequence Number: - Start / End Page: 1 - 33 Identifier: -