Linear-Size Approximations to the Vietoris–Rips Filtration

Abstract

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

Polynomial-Sized Topological Approximations Using the Permutahedron

“…These properties ensure good approximation quality and a small complex. In comparison, a cubical tessellation yields a O( √ d)-approximation of the Rips filtration which looks like an improvement over our O(d)-approximation, but the highly degenerate configuration of the cubes yields a complex size of n2 O(dk) , and therefore does not constitute an improvement over Sheehy's approach [28].…”

“…Sheehy [28] gave the first such approximate tower for Rips complexes with a formal guarantee. For 0 < ε ≤ 1/3, he constructs a (1 + ε)-approximate tower of the Rips filtration.…”

“…For 0 < ε ≤ 1/3, he constructs a (1 + ε)-approximate tower of the Rips filtration. The approximation tower of [28] consists of a filtration of simplicial complexes, whose k-skeleton has size only n(1/ε) O(λk) , where λ is the doubling dimension of the metric. Since then, several alternative techniques have been explored for Rips [12] andČech complexes [5,22], all arriving at the same complexity bound.…”

The Persistent Homology of Distance Functions under Random Projection

“…This line of research has also advanced on many different fronts. Sheehy [39] and Dey, Fan, and Wang [20] looked at sparse constructions of VietorisRips filtrations which give constant factor approximations to the persistent homology of a distance function; Oudot and Sheehy [36] used the theory of zigzag persistence to achieve a similar sparsification with strong guarantees on noise removal. Recently, Kerber and Sharathkumar [28] employed coresets for minimum enclosing balls to reduce the number of input points required.…”

PHoS: Persistent Homology for Virtual Screening

“…ing thousands of points (or even more points for low-dimensional data, if one uses a suitable approximation scheme [61,25] or an alternative construction called the alpha-filtration [26,27]). The standard algorithms use a variant of Gaussian elimination [26].…”