Local Homology Transfer and Stratification Learning

Bendich, Paul; Wang, Bei; Mukherjee, Sayan

doi:10.1137/1.9781611973099.107

Cited by 42 publications

(72 citation statements)

References 22 publications

(35 reference statements)

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“…In particular, the work in [8] focuses on studying the local structure of a sampled stratified spaces based on a multi-scale notion of local homology (see Section 2). More recent work in [9] studies how point cloud data could be clustered by strata based on how the local homology of nearby sampled points map into one another. Reconstruction and sampling.…”

Section: Introductionmentioning

confidence: 99%

Approximating Local Homology from Samples

Škraba

Wang

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult to construct in higher dimensions. We show that the persistence diagrams used for estimating local homology can be approximated using families of Vietoris-Rips complexes, whose simpler construction are robust in any dimension. To the best of our knowledge, our results, for the first time make applications based on local homology, such as stratification learning, feasible in high dimensions. IntroductionAdvances in scientific and computational experiments have improved our ability to gather large collections of data points in high-dimensional spaces. One aspect in topological data analysis is to infer the topological structure of a space given a point cloud sample. We often assume the space has manifold structure, however, more interesting cases arise when we relax our assumptions to include spaces that contains singularities and mixed dimensionality, such as stratified spaces for example.Stratified spaces can be decomposed into manifold pieces that are glued together in some uniform way. An important tool in studying these spaces is the study of the neighborhoods surrounding singularities, where manifolds of different dimensionality intersect. We focus on sampling conditions for such neighborhoods, which allow us to begin examining how successful certain reconstruction techniques are with respect to the geometric properties of the underlying shape. Our goal is to infer sampling conditions for recovering local structures of stratified spaces, in particular, the local homology groups, from a possibly noisy sampled point set. Stratification learning. In stratification learning (or mixed manifold learning), a point cloud is assumed to be sampled from a mixture of (possibly intersecting) manifolds. The objective is to recover the different

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Section: Introductionmentioning

confidence: 99%

Approximating Local Homology from Samples

Škraba

Wang

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…These element based sub-structures reveal more structure information that is directly related to the biomolecular physical, chemical, and biological properties. Persistent local homology The persistent local homology 3,9,11,12,32 is based on the algebraic topological concept called local homology groups. 49 Definition 9.…”

Section: Localized Weighted Persistent Homologymentioning

confidence: 99%

“…18-21, 51, 52, 65 Further, our LWPH models are different from traditional persistent local homology (PLH). 3,9,11,12,32,49 The PLH studies the relative homology groups between a topological space and its subspace. It is usually used to assess the local structure of a special point within a topological space.…”

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confidence: 99%

Weighted persistent homology for biomolecular data analysis

Meng

Anand

et al. 2020

Sci Rep

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In this paper, we systematically review weighted persistent homology (WPH) models and their applications in biomolecular data analysis. Essentially, the weight value, which reflects physical, chemical and biological properties, can be assigned to vertices (atom centers), edges (bonds), or higher order simplexes (cluster of atoms), depending on the biomolecular structure, function, and dynamics properties. Further, we propose the first localized weighted persistent homology (LWPH). Inspired by the great success of element specific persistent homology (ESPH), we do not treat biomolecules as an inseparable system like all previous weighted models, instead we decompose them into a series of local domains, which may be overlapped with each other. The general persistent homology or weighted persistent homology analysis is then applied on each of these local domains. In this way, functional properties, that are embedded in local structures, can be revealed. Our model has been applied to systematically studying DNA structures. It has been found that our LWPH based features can be used to successfully discriminate the A-, B-, and Z-types of DNA. More importantly, our LWPH based PCA model can identify two configurational states of DNA structure in ion liquid environment, which can be revealed only by the complicated helical coordinate system. The great consistence with the helical-coordinate model demonstrates that our model captures local structure variations so well that it is comparable with geometric models. Moreover, geometric measurements are usually defined in very local regions. For instance, the helical-coordinate system is limited to one or two basepairs. However, our LWPH can quantitatively characterize structure information in local regions or domains with arbitrary sizes and shapes, where traditional geometrical measurements fail.

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“…, with the non-trivial map having rank 1; thus L • 1 (p) manifestly has different cohomology than R [1]…”

Section: Uncovering Lower Stratamentioning

confidence: 99%

Local Cohomology and Stratification

Nanda

2019

Found Comput Math

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We outline an algorithm to recover the canonical (or, coarsest) stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a nested sequence of categories, each containing all the cells as its set of objects, with the property that two cells are isomorphic in the last category if and only if they lie in the same canonical stratum. The entire process is amenable to efficient distributed computation.

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Local Homology Transfer and Stratification Learning

Cited by 42 publications

References 22 publications

Approximating Local Homology from Samples

Approximating Local Homology from Samples

Weighted persistent homology for biomolecular data analysis

Local Cohomology and Stratification

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