62 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
Persistent Cohomology and Circular Coordinates
Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE and
Laplacian Eigenmaps address the problem of representing high-dimensional
nonlinear data in terms of low-dimensional coordinates which represent the
intrinsic structure of the data. This paradigm incorporates the assumption that
real-valued coordinates provide a rich enough class of functions to represent
the data faithfully and efficiently. On the other hand, there are simple
structures which challenge this assumption: the circle, for example, is
one-dimensional but its faithful representation requires two real coordinates.
In this work, we present a strategy for constructing circle-valued functions on
a statistical data set. We develop a machinery of persistent cohomology to
identify candidates for significant circle-structures in the data, and we use
harmonic smoothing and integration to obtain the circle-valued coordinate
functions themselves. We suggest that this enriched class of coordinate
functions permits a precise NLDR analysis of a broader range of realistic data
sets.Comment: 10 pages, 7 figures. To appear in the proceedings of the ACM
Symposium on Computational Geometry 200
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