Topological Analysis of Biological Data

Abstract: Understanding the underlying geometry and topology of biological data is a challenging problem that is key to improving inference. This new collaboration will adapt topological data analysis (TDA) tools to high-dimensional biological data, focusing on two datasets with distinct, complementary features: (i) images of morphological variation across closely related species, and (ii) transcriptomic samples of gene expression variation across related tissue samples. These very different data sets will give us a chance to study two different aspects of the usual TDA pipeline: one is the possibility of defining “testable” invariants (along the lines of property testability), i.e. ones that do not require full consideration of all the data to be approximated, and seeing whether they have utility for biological applications. The second is whether the space underlying a data set is actually Euclidean, or whether its embedding in Euclidean space (induced by the use of a number of measurements of each datum) induces metric distortion. We hope to approach this using the theory of metric distortion, and hope that this initiates a new large scale approach to understanding the geometry of data.