Persistence Barcodes Versus Kolmogorov Signatures: Detecting Modes of One-Dimensional Signals

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• 作者：Ulrich Bauer ; Axel Munk ; Hannes Sieling…
• 关键词：Persistent homology ; Mode hunting ; Exponential deviation bound ; Partial sum process ; Taut strings
• 刊名：Foundations of Computational Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：17
• 期：1
• 页码：1-33
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Economics, general; Applications of Mathematics; Linear and Multilinear Algebras, Matrix Theory; Math Applications in Computer Science; Computer Science, general;
• 出版者：Springer US
• ISSN：1615-3383
• 卷排序：17

文摘

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.