Topological Data Analysis

Abstract:

Topological Data Analysis has grown out of work focussed on deriving qualitative and yet quantifiable information about the shape of data. The underlying assumption is that knowledge of shape - the way the data are distributed in a space - permits high-level reasoning and modelling of the processes that created this data. The 0-th order aspect of shape is the number pieces: “connected components” to a topologist; “clustering” to a statistician. Higher-order topological aspects of shape are holes, quantified as “non-bounding cycles” in homology theory. These signal the existence of some type of constraint on the data- generating process.

Homology lends itself naturally to computer implementation, but its naive application is not robust to perturbations. This inspired the development of persistent homology: an algebraic topological tool that measures changes in the topology of a growing sequence of spaces (a filtration). Persistent homology provides invariants called the barcodes or persistence diagrams that are sets of intervals recording the birth and death parameter values of each homology class in the filtration. It captures information about the shape of data over a range of length scales and enables a distinction between noisy and significant structures that is continuous with respect to perturbation.  This rich geometric summary has found application in fields ranging from astrophysics to materials science and biostatistics.

Bio:

Vanessa Robins is an associate professor in ANU’s Research School of Physics. She develops theory and algorithms for the quantification of shape in data.  Her major contributions include fundamental mathematical results for persistent homology, algorithm and software development for computing topological information from digital images, and their application to the characterisation of porous and granular materials.   

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