# Improved Topological Approximations by Digitization

## TL;DR

This paper introduces an efficient approximation scheme for bech complexes that significantly reduces computational complexity for large Euclidean datasets by strategic sampling, enabling scalable topological data analysis.

## Contribution

It presents a novel bech complex approximation method with size bounds independent of data spread, improving existing schemes in Euclidean space.

## Key findings

- Achieves bech complex approximation with size (n(1/psilon))^{O(d)}
- Adds additional sample points to improve approximation accuracy
- Bound is independent of the data spread by pre-identifying critical scales

## Abstract

\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.