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 .
In this paper we propose a computationally efficient multiple hypothesis testing procedure for persistence homology. Most current approaches to hypothesis testing using persistence homology is based on resampling or permutation procedures, this is because there exists little distribution theory that provides a natural null model for persistence diagrams . In this paper we propose a null model based based approach to testing for acyclicity, coupled with a Family-Wise Error Rate (FWER) control method that does not suffer from the computational costs associated to resampling and permutation based approaches . We adapt standard False Discovery Rate (FDR) control procedures to the topological setting, and use our null model for hypothesis testing using persistent homology. A key idea in our paper is that one can compute an empirical distribution under a null hypothesis model that can be used for hypothesis testing in the same way that a normal or t-distribution is used in classical statistics. A argument for the use of this empirical null is based on simulations and limit theorems for persistent homology for point processes .
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Multiple hypothesis testing requires a control procedure
In this paper we propose a computationally efficient multiple hypothesis testing procedure for persistence homology. Most current approaches to hypothesis testing using persistence homology is based on resampling or permutation procedures, this is because there exists little distribution theory that provides a natural null model for persistence diagrams
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Simply increasing simulations or permutations to meet a Bonferroni-style threshold is prohibitively expensive.
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based
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these computational costs
the computational costs associated to resampling and permutation based approaches
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approach
procedures
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show it to be compatible both with
use
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approach and with previous approaches to hypothesis testing in
for hypothesis testing using
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By extending a limit theorem
A key idea in our paper is that one can compute an empirical distribution under a null hypothesis model that can be used for hypothesis testing in the same way that a normal or t-distribution is used in classical statistics. A argument for the use of this empirical null is based on simulations and limit theorems
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on samples from point processes, we provide theoretical validation for our FWER and FDR control methods
for point processes