\input zb-basic
\input zb-ioport
\iteman{io-port 05702494}
\itemau{Cohen-Steiner, David; Edelsbrunner, Herbert; Harer, John; Mileyko, Yuriy}
\itemti{Lipschitz functions have $L_{p}$-stable persistence.}
\itemso{Found. Comput. Math. 10, No. 2, 127-139 (2010).}
\itemab
The results in this paper are motivated by the analysis of gene expression data during development. The biological data consists of a series of 13 measurements taken over the development stages of 14 cell types. For each cell type the 13 measurements can be viewed as a samples of a function $f:[0,1] \rightarrow {\mathbb R}$. The underlying function is assumed to be Lipschitz. Each cross section represents a level surface. For the general results the authors consider a Lipschitz function $f : {\mathbb R} \rightarrow {\mathbb R}$. Let ${\mathbb X}_{a} = f^{-1}(-\infty,a]$ be the subset defined by a threshold $a$. Then as $a$ increases the homology groups of $ {\mathbb X}_{a}$ change. Thus one can observe homology classes appear and disappear. For a given homology class $\alpha$ one can define a point in $(d(\alpha), b(\alpha)) \in {\mathbb R}^{2}$, where $b(\alpha)$ is the birth level and $d(\alpha)$ is the death level. For a given $\ell$ the set $Dgm_{\ell}(f)$ of these points corresponding to $\ell$-dimensional homology classes is called the $\ell$-th persistence diagram of $f$. The Wasserstein distance between the diagrams for two functions $f$ and $g$ gives a measure of the (dis)-similarity between the two functions. The authors show a stability result for two Lipschitz functions on triangulable metric spaces that bounds the Wasserstein distance between them in terms of the $L_{\infty}$ distance between them.
\itemrv{Jonathan Hodgson (Philadelphia)}
\itemcc{}
\itemut{Wasserstein distance; persistent homology; gene expression; Lipschitz function}
\itemli{doi:10.1007/s10208-010-9060-6}
\end