Persistence-based clustering in riemannian manifolds

Chazal, Frédéric; Guibas, Leonidas J.; Oudot, Steve; Škraba, Primož

doi:10.1145/1998196.1998212

Cited by 56 publications

(52 citation statements)

References 25 publications

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“…Note that one could also compute the structural stability margin using Morse's Lemma, or the statistics of the detector (e.g., the second-moment matrix). Finally, the literature on Persistent Topology [25,26,27] also provides methods to quantify the life-span of structures, which can be used as a proxy of structural stability margin. Indeed, the notion of structural stability proposed above is a special case of persistent topology.…”

Section: Designing Feature Detectorsmentioning

confidence: 99%

Video-based descriptors for object recognition

Lee

Soatto

2011

Image and Vision Computing

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Section: Designing Feature Detectorsmentioning

confidence: 99%

Video-based descriptors for object recognition

Lee

Soatto

2011

Image and Vision Computing

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“…74 The method clusters the high-dimensional data set according to the spatial density function and merges the inconsequential clusters into noise. In the spatial density functions, the height of the peak is the density of the basin, and the position of the peak is the candidate cluster center.…”

Section: Locally Scaled Diffusion Mapmentioning

confidence: 99%

A comparative analysis of clustering algorithms: O2 migration in truncated hemoglobin I from transition networks

Cazade

Zheng

Prada‐Gracia

et al. 2015

The Journal of Chemical Physics

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The ligand migration network for O2–diffusion in truncated Hemoglobin N is analyzed based on three different clustering schemes. For coordinate-based clustering, the conventional k–means and the kinetics-based Markov Clustering (MCL) methods are employed, whereas the locally scaled diffusion map (LSDMap) method is a collective-variable-based approach. It is found that all three methods agree well in their geometrical definition of the most important docking site, and all experimentally known docking sites are recovered by all three methods. Also, for most of the states, their population coincides quite favourably, whereas the kinetics of and between the states differs. One of the major differences between k–means and MCL clustering on the one hand and LSDMap on the other is that the latter finds one large primary cluster containing the Xe1a, IS1, and ENT states. This is related to the fact that the motion within the state occurs on similar time scales, whereas structurally the state is found to be quite diverse. In agreement with previous explicit atomistic simulations, the Xe3 pocket is found to be a highly dynamical site which points to its potential role as a hub in the network. This is also highlighted in the fact that LSDMap cannot identify this state. First passage time distributions from MCL clusterings using a one- (ligand-position) and two-dimensional (ligand-position and protein-structure) descriptor suggest that ligand- and protein-motions are coupled. The benefits and drawbacks of the three methods are discussed in a comparative fashion and highlight that depending on the questions at hand the best-performing method for a particular data set may differ.

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“…Hence one may also ask what the 0-th persistent homology of R f (X) induced by f is. This turns out to be the same as approximating the 0-th persistent homology for X and can be solved using results from [3,4]. Therefore, the only remaining issue is to approximate the 1-st homology of a Reeb graph.…”

Section: Remarkmentioning

confidence: 99%

Reeb Graphs: Approximation and Persistence

Dey

Wang

2012

Discrete Comput Geom

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Given a continuous function f : X → IR on a topological space X, its level set f −1 (a) changes continuously as the real value a changes. Consequently, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f summarizes this information into a graph structure. Previous work on Reeb graph mainly focused on its efficient computation. In this paper, we initiate the study of two important aspects of the Reeb graph which can facilitate its broader applications in shape and data analysis.The first one is the approximation of the Reeb graph of a function on a smooth compact manifold M without boundary. The approximation is computed from a set of points P sampled from M. By leveraging a relation between the Reeb graph and the so-called vertical homology group, as well as between cycles in M and in a Rips complex constructed from P , we compute the H 1 -homology of the Reeb graph from P . It takes O(n log n) expected time, where n is the size of the 2-skeleton of the Rips complex. As a by-product, when M is an orientable 2-manifold, we also obtain an efficient near-linear time (expected) algorithm for computing the rank of H 1 (M) from point data. The best known previous algorithm for this problem takes O(n 3 ) time for point data.The second aspect concerns the definition and computation of the persistent Reeb graph homology for a sequence of Reeb graphs defined on a filtered space. For a piecewise-linear function defined on a filtration of a simplicial complex K, our algorithm computes all persistent H 1 -homology for the Reeb graphs in O(nn 3 e ) time, where n is the size of the 2-skeleton and n e is the number of edges in K.

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Persistence-based clustering in riemannian manifolds

Cited by 56 publications

References 25 publications

Video-based descriptors for object recognition

Video-based descriptors for object recognition

A comparative analysis of clustering algorithms: O2 migration in truncated hemoglobin I from transition networks

Reeb Graphs: Approximation and Persistence

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