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Improved Topological Approximations by Digitization by Choudhary, Aruni; Kerber, Michael; Raghvendra, Sharath is a scholarly article available to read on EtoBox.

What is Improved Topological Approximations by Digitization about?

\v{C}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 $(1+\epsilon)$-approximating the topological information of the \v{C}ech complexes for $n$ points in $\mathbb{R}^d$, for $\epsilon\in(0,1]$. Our approximation has a total size of $n\left(\frac{1}{\epsilon}\right)^{O(d)}$ for constant dimension $d$, improving all the currently available $(1+\epsilon)$-approximation schemes of simplicial filtrations in Euclidean space. Perhaps counter-intuitively, we arrive at our result by adding additional $n\left(\frac{1}{\epsilon}\right)^{O(d)}$ 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 \v{C}ech complexes changes and sampling accordingly.

Author

Choudhary, Aruni; Kerber, Michael; Raghvendra, Sharath

Published

2018

Language

EN