Efficient and Robust Persistent Homology for Measures

Buchet, Mickaël; Chazal, Frédéric; Oudot, Steve; Sheehy, Donald R.

doi:10.1137/1.9781611973730.13

Cited by 31 publications

(25 citation statements)

References 17 publications

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“…The essential idea of weighted persistent homology models is to introduce a specially-designed weight function/parameter that incorporates the biomolecular physical, chemical or biological properties, into the construction of simplicial complexes or homology generator evaluation. Generally speaking, all these WPH models can be classified into three types, including vertex-weighted, 8,17,29,36,69,71 edge-weighted, 14,18,21,55,68,69 and simplex-weighted models. 25,56,64 To facilitate our discussion, we define a weighted point set as (X, V ) with X = {x i | i=1,2,...,N } and V = {v i | i=1,2,...,N }.…”

Section: Weighted Persistent Homologymentioning

confidence: 99%

“…Generally speaking, the weighted persistent homology can be characterized into three major categories, vertex-weighted, 8,17,29,36,69,71 edge-weighted, 14,18,21,55,68,69 and simplex-weighted models. 25,56,64 For vertexweighted models, a weight value is defined on each vertex.…”

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confidence: 99%

“…Other vertex-weighted models consider different types of weighted distance functions. 8,17,36 In a weighted Vietoris-Rips andČech complex model, a new distance function is proposed as a minimal value of the scaled Euclidean distances between the position to all atoms. 8 The inverse of the distance function represents the union of balls centered at the atom, and naturally induces weightedČech complexex through nerve theorem.…”

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confidence: 99%

“…14,55 Using weight value as a filtration parameter, Vietoris-Rips or clique complex can be defined as a maximal simplicial complex, whose 1-skeleton has weight values larger (or smaller) than a certain filtration value. 17 For weighted clique rank homology model, 55 a network/graph, with a weight value on each edge, is considered. Clique complex can be defined on the subgraph composed of edges with weight larger than filtration value.…”

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confidence: 99%

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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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Section: Weighted Persistent Homologymentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Weighted persistent homology for biomolecular data analysis

Meng

Anand

et al. 2020

Sci Rep

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“…Assume that (X, ρ) is a metric space and let X ⊆ X. Singlelinkage hierarchical clustering problems based on X are naturally associated with the function f : X → [0, ∞) given by PD(V Z ) Figure 1: A schematic diagram illustrating the potential locations of persistence points of PD(V ) based on the computation of PD(V Z ). The persistence point (2,6) of PD(V Z ) is matched with a persistence point of PD(V ) which must lie in the light gray region. If a persistence point of PD(V Z ) occurs at one of the open circles, then it is possible that it is a computational artifact, i.e.…”

Section: Introductionmentioning

confidence: 99%

A comparison framework for interleaved persistence modules

Harker

Kramar

Levanger

et al. 2019

J Appl. and Comput. Topology

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We present a generalization of the induced matching theorem of [1] and use it to prove a generalization of the algebraic stability theorem for R-indexed pointwise finitedimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance. 1 for choices of t ∈ [0, ∞). The collection {C(f, t)} t∈R is called the sublevel set filtration of X induced by f . Superlevel sets and superlevel set filtrations are defined similarly by considering the sets {x ∈ X : t ≤ f (x)} for every t ∈ R.Alternatively, assume that X is a topological domain and f : X → R is a scalar value of a nonlinear physical model, e.g. the magnitude of vorticity or temperature field of a fluid, the chemical density in a reaction diffusion system, the magnitude of forces between particles in a granular system, etc. Patterns produced by these systems are often associated with sublevel or superlevel sets of f . In fact, the direct motivation for this work is to justify claims made in [16] concerning the time-evolution of patterns in convection models.These examples are meant to motivate our interest in studying the geometry of the sets C(f, t). Homology provides a coarse but computable representation of this geometry. In particular, for each t ∈ R, there is an assigned graded vector space, the inclusion maps induce the following linear maps at the level of homology:This homological information can be abstracted as follows.Definition 1.1. A persistence module is a collection of vector spaces indexed by the real numbers, {V t } t∈R , and linear maps {ϕ V (s, t) : V s → V t } s≤t∈R satisfying the following conditions:We write (V, ϕ V ) to denote the collection of vector spaces and compatible linear maps, and will sometimes just write V for the full persistence module when the maps are clear. We say that V is a pointwise finite dimensional (PFD) persistence module when every V t is finite-dimensional.As is described in Sections 2.1 and 4, a PFD persistence module gives rise to a persistence diagram, which is a set of points inGiven a PFD persistence module (V, ϕ V ), we denote its associated persistence diagram by PD(V ). Observe that we have outlined a procedure by which the sublevel sets of a scalar field f produce a persistence diagram PD. Returning to our examples, in the first case, it is reasonable to assume that the actual available data is X ′ ⊆ X ⊆ X, as opposed to X , which represents the true set of objects upon which the clustering is to be based. In this case, collecting experimental or numerical data results in f ′ : X → R, an approximation of the actual function of interest, f . Recent computational developments have led to the routine computation of PD ′ , the persistence diagram associated with X ′ or f ′ . Thus, the natural question is this: how is PD ′ , the computed persistence diagram, related to PD, the persistence diagr...

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DTM-Based Filtrations

Anai

Chazal²,

Glisse³

et al. 2020

Topological Data Analysis

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Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. theČech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper, we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions, and extends some previous work on the approximation of such functions.

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Efficient and Robust Persistent Homology for Measures

Cited by 31 publications

References 17 publications

Weighted persistent homology for biomolecular data analysis

Weighted persistent homology for biomolecular data analysis

A comparison framework for interleaved persistence modules

DTM-Based Filtrations

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