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Statistical aspects of persistent homology
Fri 03 May 2013, 16:30
Matthew Arnold
Bristol
Organisers: Nick Whiteley, Feng Yu
ABSTRACT
We are studying statistical aspects of persistent homology, within the framework of "Topological Data Analysis". For points sampled uniformly at random from a topological object in Euclidean space, the persistence barcodes contain some intervals that represent topological "features" and some intervals that represent "noise" which occurs as a result of our uniform sampling strategy. We investigate the null distribution of noisy intervals for uniform samples, by investigating the distribution of the full barcode for a uniform sample from the unit square. This directs us towards algorithmically identifying noisy intervals in a given barcode, so that the remaining intervals are classified as actual features of the underlying space. Related to this is the study of how the noisy intervals change when the sampled points are not uniformly sampled from the object. By using a kernel density estimate of the sample we can correct for the non-uniformity to return barcodes that behave similarly to those we would expect if we could have sampled uniformly. This allows us to identify the noisy intervals using the null distribution described earlier. Some ideas in both of these areas will be presented.