Improved Topological Approximations by Digitization

Aruni Choudhary, Michael Kerber, Sharath Raghvendra

Abstract: ech complexes are useful simplicial complexes for computing and analyzing topological features of data that lies in Euclidean space. Unfortunately, computing these complexes becomes prohibitively expensive for large-sized data sets even for medium-to-low dimensional data. We present an approximation scheme for (1 + ε)-approximating the topological information of the Čech complexes for n points in ℝd, for ε ∊ (0, 1]. Our approximation has a total size of for constant dimension d, improving all the currently available (1 + ε)-approximation schemes of simplicial filtrations in Euclidean space. Perhaps counter-intuitively, we arrive at our result by adding additional sample points to the input. We achieve a bound that is independent of the spread of the point set by pre-identifying the scales at which the Čech complexes changes and sampling accordingly.