The Persistent Homology of Distance Functions under Random Projection

Sheehy, Donald R.

doi:10.1145/2582112.2582126

Cited by 25 publications

(35 citation statements)

References 32 publications

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“…To adapt the aforementioned schemes to play nice with high dimensional point clouds, it makes sense to use dimension reduction results to eliminate the dependence on λ. Indeed, it has been shown, in analogy to the famous Johnson-Lindenstrauss Lemma [18], that an orthogonal projection of a point set of R d to a O(log n/ε 2 )-dimensional subspace yields a (1 + ε) approximation of theČech filtration [20,29]. Combining these two approximation schemes, however, yields an approximation of size O(n k+1 ) (ignoring ε-factors) and does not improve upon the exact case.…”

Section: Motivation and Previous Workmentioning

confidence: 99%

Polynomial-Sized Topological Approximations Using the Permutahedron

Choudhary

Kerber

Raghvendra

2017

Discrete Comput Geom

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Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in R d , we obtain a O(d)-approximation whose k-skeleton has size n2 O(d log k) per scale and n2 O(d log d) in total over all scales. In conjunction with dimension reduction techniques, our approach yields a O(polylog(n))-approximation of size n O(1) for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry. Building on the same geometric concept, we also present a lower bound result on the size of an approximation: we construct a point set for which every (1 + ε)-approximation of theČech filtration has to contain n (log log n) features, provided that ε < 1 log 1+c n for c ∈ (0, 1).

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Section: Motivation and Previous Workmentioning

confidence: 99%

Polynomial-Sized Topological Approximations Using the Permutahedron

Choudhary

Kerber

Raghvendra

2017

Discrete Comput Geom

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“…Theorem 1.1 is based on, and recovers as a special case, an extension of the Johnson-Lindenstrauss theorem by Sheehy [6] (see the example with the Gaussian width of discrete sets below). One crucial difference to the classical approach is that the Gaussian width allows us to do better when the data X has a particularly simple structure.…”

Section: Introductionmentioning

confidence: 95%

Persistent homology for low-complexity models

Lotz

2019

Proc. R. Soc. A.

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We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is determined by the Gaussian width of a structure associated to the data set, rather than its size, and such a reduction can be computed efficiently. We further relate the Gaussian width to the doubling dimension of a finite metric space, which appears in the study of the complexity of other methods for approximating persistent homology. This allows to literally replace the ambient dimension by an intrinsic notion of dimension related to the structure of the data.

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“…Roughly, the bottleneck distance d b (B, C) between two barcodes B and C is the magnitude of a perturbation of B required to transform B into C (or vice versa); here, magnitude is defined as the maximum distance an endpoint of any interval moves in the perturbation [21]. This distance has very good theoretical properties, and plays a important role in the theoretical foundations of topological data analysis [20,18,9,62,30]. The bottleneck distance is also readily computed [42], and together with its variant the Wasserstein distance, is commonly used in applications [29,2].…”

Section: Metrics On Topological Signatures Of Bifiltrationsmentioning

confidence: 99%

Persistent Homology for Virtual Screening

Keller¹,

Lesnick²,

Willke³

2018

Preprint

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<div> <div> <div> <p>Finding new medicines is one of the most important tasks of pharmaceutical companies. One of the best approaches to finding a new drug starts with answering this simple question: Given a known effective drug X, what are the top 100 molecules in our database most similar to X? Thus the essence of the problem is a nearest-neighbors search, and the key question is how to define the distance between two molecules in the database. In this paper, we investigate the use of topological, rather than geometric, or chemical, signatures for molecules, and two notions of distance that come from comparing these topological signatures. We introduce PH_VS (Persistent Homology for Virtual Screening), a new system for ligand-based screening using a topological technique known as multi-parameter persistent homology. We show that our approach can match or exceed a reasonable estimate of current state of the art (including well-funded commercial tools), even with relatively little domain-specific tuning. Indeed, most of the components we have built for this system are general-purpose tools for data science and will be released soon as open source software. </p> </div> </div> </div>

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The Persistent Homology of Distance Functions under Random Projection

Cited by 25 publications

References 32 publications

Polynomial-Sized Topological Approximations Using the Permutahedron

Polynomial-Sized Topological Approximations Using the Permutahedron

Persistent homology for low-complexity models

Persistent Homology for Virtual Screening

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