10,610 research outputs found
Multiple testing with persistent homology
Multiple hypothesis testing requires a control procedure. Simply increasing
simulations or permutations to meet a Bonferroni-style threshold is
prohibitively expensive. In this paper we propose a null model based approach
to testing for acyclicity, coupled with a Family-Wise Error Rate (FWER) control
method that does not suffer from these computational costs. We adapt an False
Discovery Rate (FDR) control approach to the topological setting, and show it
to be compatible both with our null model approach and with previous approaches
to hypothesis testing in persistent homology. By extending a limit theorem for
persistent homology on samples from point processes, we provide theoretical
validation for our FWER and FDR control methods
Topological analysis of scalar fields with outliers
Given a real-valued function defined over a manifold embedded in
, we are interested in recovering structural information about
from the sole information of its values on a finite sample . Existing
methods provide approximation to the persistence diagram of when geometric
noise and functional noise are bounded. However, they fail in the presence of
aberrant values, also called outliers, both in theory and practice.
We propose a new algorithm that deals with outliers. We handle aberrant
functional values with a method inspired from the k-nearest neighbors
regression and the local median filtering, while the geometric outliers are
handled using the distance to a measure. Combined with topological results on
nested filtrations, our algorithm performs robust topological analysis of
scalar fields in a wider range of noise models than handled by current methods.
We provide theoretical guarantees and experimental results on the quality of
our approximation of the sampled scalar field
Optimal rates of convergence for persistence diagrams in Topological Data Analysis
Computational topology has recently known an important development toward
data analysis, giving birth to the field of topological data analysis.
Topological persistence, or persistent homology, appears as a fundamental tool
in this field. In this paper, we study topological persistence in general
metric spaces, with a statistical approach. We show that the use of persistent
homology can be naturally considered in general statistical frameworks and
persistence diagrams can be used as statistics with interesting convergence
properties. Some numerical experiments are performed in various contexts to
illustrate our results
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