Sketching Persistence Diagrams, Don Sheehy

30 April 2021
15:00
Don Sheehy

Further Information: 

Don Sheehy is an Associate Professor of Computer Science at North Carolina State University.  He received his B.S.E. from Princeton University and his Ph.D. in Computer Science from Carnegie Mellon University.   He spent two years as a postdoc at Inria Saclay in France.  His research is in algorithms and data structures in computational geometry and topological data analysis.  

Abstract

Given a persistence diagram with n points, we give an algorithm that produces a sequence of n persistence diagrams converging in bottleneck distance to the input diagram, the ith of which has i distinct (weighted) points and is a 2-approximation to the closest persistence diagram with that many distinct points. For each approximation, we precompute the optimal matching between the ith and the (i+1)st. Perhaps surprisingly, the entire sequence of diagrams as well as the sequence of matchings can be represented in O(n) space. The main approach is to use a variation of the greedy permutation of the persistence diagram to give good Hausdorff approximations and assign weights to these subsets. We give a new algorithm to efficiently compute this permutation, despite the high implicit dimension of points in a persistence diagram due to the effect of the diagonal. The sketches are also structured to permit fast (linear time) approximations to the Hausdorff distance between diagrams -- a lower bound on the bottleneck distance. For approximating the bottleneck distance, sketches can also be used to compute a linear-size neighborhood graph directly, obviating the need for geometric data structures used in state-of-the-art methods for bottleneck computation.

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  • Applied Topology Seminar