In an era of increasing digitization, the data we can collect and analyze has become increasingly complex. In particular we are wanting methods to make sense of data where each object itself has interesting structure or shape - for example we have a collection of graphs or a collection of scanned geometric objects. Topological Data Analysis provides methods to create a topological signature of each objectsummary that captures topological and geometric information. Because these topological signatures lie in a common space, we can statistically analyse these topological signatures. This can be both more tractable than working with the raw data and also can identify relevant global and local features.
One facet of my reaserch is statistical theory for working with these topological signatures - and in particular statistical theory about persistence diagrams.
Some papers about statistical theory in TDA:
Fréchet means for distributions of persistence diagrams
K Turner, Y Mileyko, S Mukherjee, J Harer - Discrete & Computational Geometry, 2014
Persistence diagrams are common objects in topological data analysis but the space of persistence diagrams is complicated making statistical analysis challenging. Previously the mean of a set of persistence diagrams was known to exist but was not understood. This paper characterises the mean and gave an algorithm to compute it.
V Robins, K Turner - Physica D: Nonlinear Phenomena, 2016
Principal component analysis (PCA) is a standard tool in data analysis. This paper adapts and translate PCA for topological data analysis by constructing rank functions which lie in a linear space. We then demonstrate the exploratory and explanatory power of the PCA of rank functions in variety of case studies.
Wasserstein stability for persistence diagrams
P Skraba, K Turner - arXiv preprint arXiv:2006.16824, 2020
Persistence diagrams are often compare using the Wasserstein distance. Errors and rounding in data collection, alongside natural variation, cause perturbations of persistence diagrams. This paper bounds the effect of this randomness in the Wasserstein distance. The paper also develops the algebraic framework for Wasserstein distances impacting theory of multi-parameter persistence.
Probabilistic Fréchet means for time varying persistence diagrams
E Munch, K Turner, P Bendich, S Mukherjee, J Mattingly, J Harer - Electronic Journal of Statistics, 2015
Hypothesis testing for topological data analysis
A Robinson, K Turner - Journal of Applied and Computational Topology, 2017
Same but different: distance correlations between topological summaries
K Turner, G Spreemann - Topological Data Analysis, 2020
Intrinsic interleaving distance for merge trees
E Gasparovic, E Munch, S Oudot, K Turner, B Wang, Y Wang - arXiv preprint arXiv:1908.00063, 2019
Means and medians of sets of persistence diagrams
K Turner - Homology, Homotopy and Applications, 2020