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The Vietoris-Rips filtration is a versatile tool in topological dataanalysis. It is a sequence of simplicial complexes built on a metric space toadd topological structure to an otherwise disconnected set of points. It iswidely used because it encodes useful information about the topology of theunderlying metric space. This information is often extracted from its so-calledpersistence diagram. Unfortunately, this filtration is often too large toconstruct in full. We show how to construct an O(n)-size filtered simplicialcomplex on an $n$-point metric space such that its persistence diagram is agood approximation to that of the Vietoris-Rips filtration. This new filtrationcan be constructed in $O(n\log n)$ time. The constant factors in both the sizeand the running time depend only on the doubling dimension of the metric spaceand the desired tightness of the approximation. For the first time, this makesit computationally tractable to approximate the persistence diagram of theVietoris-Rips filtration across all scales for large data sets.  We describe two different sparse filtrations. The first is a zigzagfiltration that removes points as the scale increases. The second is a(non-zigzag) filtration that yields the same persistence diagram. Both methodsare based on a hierarchical net-tree and yield the same guarantees.

Linear-Size Approximations to the Vietoris-Rips Filtration