🍩 Database of Original & Non-Theoretical Uses of Topology
(found 3 matches in 0.001237s)
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Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis (2015)
Jose A. Perea, John HarerAbstract
We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS, which stands for Sliding Windows and 1-Dimensional Persistence Scoring. -
Lipschitz Functions Have Lp-Stable Persistence (2010)
David Cohen-Steiner, Herbert Edelsbrunner, John Harer, Yuriy MileykoAbstract
We prove two stability results for Lipschitz functions on triangulable, compact metric spaces and consider applications of both to problems in systems biology. Given two functions, the first result is formulated in terms of the Wasserstein distance between their persistence diagrams and the second in terms of their total persistence.