TY - GEN
T1 - The density of expected persistence diagrams and its kernel based estimation
AU - Chazal, Frédéric
AU - Divol, Vincent
N1 - Publisher Copyright:
© Frédéric Chazal and Vincent Divol; licensed under Creative Commons License CC-BY 34th Symposium on Computational Geometry (SoCG 2018).
PY - 2018/6/1
Y1 - 2018/6/1
N2 - Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane R2 that can equivalently be seen as discrete measures in R2. When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the Čech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on R2, has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [1] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density.
AB - Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane R2 that can equivalently be seen as discrete measures in R2. When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the Čech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on R2, has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [1] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density.
KW - Persistence diagrams
KW - Subanalytic geometry
KW - Topological data analysis
U2 - 10.4230/LIPIcs.SoCG.2018.26
DO - 10.4230/LIPIcs.SoCG.2018.26
M3 - Conference contribution
AN - SCOPUS:85048980527
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 261
EP - 2615
BT - 34th International Symposium on Computational Geometry, SoCG 2018
A2 - Toth, Csaba D.
A2 - Speckmann, Bettina
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 34th International Symposium on Computational Geometry, SoCG 2018
Y2 - 11 June 2018 through 14 June 2018
ER -