Linear-size approximations to the vietoris-rips filtration

Sheehy, Donald R.

doi:10.1145/2261250.2261286

Cited by 52 publications

(96 citation statements)

References 35 publications

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“…Note however that the exact complex size is bigger than the one achieved with the other types of Rips zigzags when ζ < 1. These asymptotic bounds are as good in order of magnitude as the ones achieved with previous lightweight structures [7,16].…”

Section: Complexity Boundssupporting

confidence: 56%

“…Recently, Sheehy [16] proposed a method for building a sparse zigzag filtration whose barcode is provably close to that of the Rips filtration as well as a non-zigzagging variant achieving similar guarantees. Also, Dey et al gave an alternative persistence algorithm for simplicial maps rather than inclusions, which is closely related to zigzag persistence [7].…”

Section: Introductionmentioning

confidence: 99%

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Zigzag zoology

Oudot

Sheehy

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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For points sampled near a compact set X, the persistence barcode of the Rips filtration built from the sample contains information about the homology of X as long as X satisfies some geometric assumptions. The Rips filtration is prohibitively large, however zigzag persistence can be used to keep the size linear. We present several species of Ripslike zigzags and compare them with respect to the signalto-noise ratio, a measure of how well the underlying homology is represented in the persistence barcode relative to the noise in the barcode at the relevant scales. Some of these Rips-like zigzags have been available as part of the Dionysus library for several years while others are new. Interestingly, we show that some species of Rips zigzags will exhibit less noise than the (non-zigzag) Rips filtration itself. Thus, Rips zigzags can offer improvements in both size complexity and signal-to-noise ratio.Along the way, we develop new techniques for manipulating and comparing persistence barcodes from zigzag modules. We give methods for reversing arrows and removing spaces from a zigzag while controlling the changes occurring in its barcode. We also discuss factoring zigzags and a kind of interleaving of two zigzags that allows their barcodes to be compared. These techniques were developed to provide our theoretical analysis of the signal-to-noise ratio of Rips-like zigzags, but they are of independent interest as they apply to zigzag modules generally.

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Section: Complexity Boundssupporting

confidence: 56%

Section: Introductionmentioning

confidence: 99%

Zigzag zoology

Oudot

Sheehy

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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“…It retains the simplicity of the Rips complex as well as the sparsity of the witness complex. Its construction resembles the sparsified Rips complex proposed in [23] and also the combinatorial Delaunay triangulation proposed in [6], but it does not build a Rips complex on the subsample and thus is sparser than the Rips complex with the same set of vertices. This fact makes a real difference in practice as our preliminary experiments show.…”

Section: Introductionmentioning

confidence: 99%

Graph induced complex on point data

Dey

Fan

Wang

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the most favored complexes for this purpose have so far been Vietoris-Rips and witness complexes. While the Vietoris-Rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces, its size is hugeparticularly in high dimensions. The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structures. We investigate a complex called the graph induced complex that, to some extent, enjoys the advantages of both. It works on a subsample but still retains the power of capturing the topology as the Vietoris-Rips complex. It only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably. We show that, using the graph induced complex one can (i) infer the one-dimensional homology of a manifold from a very lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample without computing Delaunay triangulations, (iii) infer the persistent homology groups of compact sets from a sufficiently dense sample. We provide experimental evidences in support of our theory.

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“…Topological Approximation. Computational infeasibility of theČech complex and Vietoris-Rips complex motivates the development of approximate simplicial representations such as the lazy witness complexes, sparse-Rips complex [26] and graph induced complex (GIC) [10]. Sparse-Rips complex [26] perturbs the distance metric in such a way that when the regions covered by a point can be covered by its neighbouring points, that point can be deleted without changing the topology.…”

Section: Computation Of Topological Featuresmentioning

confidence: 99%

“…A simplicial representation facilitates computation of basic topological objects such as simplicial complexes, filtrations, and persistent homologies. Thus, researchers devised approximations of theČech complex as well as its best possible approximation the Vietoris-Rips complex [7,26,10]. One of such computationally feasible arXiv:1906.06122v2 [cs.CG] 19 Jun 2019 approximate simplicial representation is the lazy witness complex [7].…”

Section: Introductionmentioning

confidence: 99%

Topological Data Analysis with $$\epsilon $$-net Induced Lazy Witness Complex

Basu

Bressan

2019

Lecture Notes in Computer Science

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Topological data analysis computes and analyses topological features of the point clouds by constructing and studying a simplicial representation of the underlying topological structure. The enthusiasm that followed the initial successes of topological data analysis was curbed by the computational cost of constructing such simplicial representations. The lazy witness complex is a computationally feasible approximation of the underlying topological structure of a point cloud. It is built in reference to a subset of points, called landmarks, rather than considering all the points as in theČech and Vietoris-Rips complexes. The choice and the number of landmarks dictate the effectiveness and efficiency of the approximation. We adopt the notion of -cover to define -net. We prove that -net, as a choice of landmarks, is an -approximate representation of the point cloud and the induced lazy witness complex is a 3-approximation of the induced Vietoris-Rips complex. Furthermore, we propose three algorithms to construct -net landmarks. We establish the relationship of these algorithms with the existing landmark selection algorithms. We empirically validate our theoretical claims. We empirically and comparatively evaluate the effectiveness, efficiency, and stability of the proposed algorithms on synthetic and real datasets.

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Linear-size approximations to the vietoris-rips filtration

Cited by 52 publications

References 35 publications

Zigzag zoology

Zigzag zoology

Graph induced complex on point data

Topological Data Analysis with $$\epsilon $$-net Induced Lazy Witness Complex

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