Fréchet Means for Distributions of Persistence Diagrams

Turner, Katharine; Mileyko, Yuriy; Mukherjee, Sayan; Harer, John

doi:10.1007/s00454-014-9604-7

Cited by 217 publications

(274 citation statements)

References 26 publications

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“…The mean and variance appropriate for the space are the so-called Fréchet mean and variance. The Fréchet mean, however, is not unique (Turner et al, 2014), rendering it a challenging statistical issue to perform inference on PDs directly. Notable PD applications in imaging studies also realize that inference on PD is by no means straightforward in practice (Chung, Bubenik and Kim, 2009; Gamble and Heo, 2010; Heo, Gamble and Kim, 2012).…”

Section: Methodsmentioning

confidence: 99%

Topological data analysis of single-trial electroencephalographic signals

Wang¹,

Ombao²,

Chung³

2018

Ann. Appl. Stat.

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Epilepsy is a neurological disorder that can negatively affect the visual, audial and motor functions of the human brain. Statistical analysis of neurophysiological recordings, such as electroencephalogram (EEG), facilitates the understanding and diagnosis of epileptic seizures. Standard statistical methods, however, do not account for topological features embedded in EEG signals. In the current study, we propose a persistent homology (PH) procedure to analyze single-trial EEG signals. The procedure denoises signals with a weighted Fourier series (WFS), and tests for topological difference between the denoised signals with a permutation test based on their PH features persistence landscapes (PL). Simulation studies show that the test effectively identifies topological difference and invariance between two signals. In an application to a single-trial multichannel seizure EEG dataset, our proposed PH procedure was able to identify the left temporal region to consistently show topological invariance, suggesting that the PH features of the Fourier decomposition during seizure is similar to the process before seizure. This finding is important because it could not be identified from a mere visual inspection of the EEG data and was in fact missed by earlier analyses of the same dataset.

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Section: Methodsmentioning

confidence: 99%

Topological data analysis of single-trial electroencephalographic signals

Wang¹,

Ombao²,

Chung³

2018

Ann. Appl. Stat.

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“…To address this issue, a promising alternative consists in considering the barycenter of a set of diagrams, given a distance metric between them, such as the so-called Wasserstein metric [27], hence the term Wasserstein barycenter. For this, an algorithm has been proposed by Turner et al [94]. However, it is based on an iterative procedure, for which each iteration relies itself on a demanding optimization problem (optimal assignment in a weighted bipartite graph [64]), which makes it impractical for real-life datasets.…”

Section: Introductionmentioning

confidence: 99%

“…Such representative diagrams are obtained by computing explicitly the discrete Wasserstein barycenter of the set of persistence diagrams, a notoriously computationally intensive task. In particular, we revisit efficient algorithms for Wasserstein distance approximation [12,51] to extend previous work on barycenter estimation [94]. We present a new fast algorithm, which progressively approximates the barycenter by iteratively increasing the computation accuracy as well as the number of persistent features in the output diagram.…”

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confidence: 99%

Progressive Wasserstein Barycenters of Persistence Diagrams

Vidal

Budin

Tierny

2019

IEEE Trans. Visual. Comput. Graphics

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Fig. 1. The Persistence diagrams of three members (a-c) of the Isabel ensemble (wind velocity) concisely and visually encode the number, data range and salience of the features of interest found in the data (eyewall and region of high speed wind, blue and red in (a)). In these diagrams, features with a persistence smaller than 10% of the function range or on the boundary are shown in transparent white. The pointwise mean for these three members (d) exhibits three salient interior features (due to distinct eyewall locations, blue, green and red), although the diagrams of the input members only report two salient interior features at most, located at drastically different data ranges (the red feature is further down the diagonal in (a) and (b)). The Wasserstein barycenter of these three diagrams (e) provides a more representative view of the features found in this ensemble, as it reports a feature number, range and salience that better matches the input diagrams (a-c). Our work introduces a progressive approximation algorithm for such barycenters, with fast practical convergence. Our framework supports computation time constraints (e) which enables the approximation of Wasserstein barycenters within interactive times. We present an application to the clustering of ensemble members based on their persistence diagrams ((f), lifting: α = 0.2), which enables the visual exploration of the main trends of features of interest found in the ensemble.Abstract-This paper presents an efficient algorithm for the progressive approximation of Wasserstein barycenters of persistence diagrams, with applications to the visual analysis of ensemble data. Given a set of scalar fields, our approach enables the computation of a persistence diagram which is representative of the set, and which visually conveys the number, data ranges and saliences of the main features of interest found in the set. Such representative diagrams are obtained by computing explicitly the discrete Wasserstein barycenter of the set of persistence diagrams, a notoriously computationally intensive task. In particular, we revisit efficient algorithms for Wasserstein distance approximation [12,51] to extend previous work on barycenter estimation [94]. We present a new fast algorithm, which progressively approximates the barycenter by iteratively increasing the computation accuracy as well as the number of persistent features in the output diagram. Such a progressivity drastically improves convergence in practice and allows to design an interruptible algorithm, capable of respecting computation time constraints. This enables the approximation of Wasserstein barycenters within interactive times. We present an application to ensemble clustering where we revisit the k-means algorithm to exploit our barycenters and compute, within execution time constraints, meaningful clusters of ensemble data along with their barycenter diagram. Extensive experiments on synthetic and real-life data sets report that our algorithm converges to barycenters that are qualitatively meaningfu...

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“…Recent applications of persistence often confront dynamically evolving data [2,17], and in these cases one requires the ability to make inferences about the dynamics from collections of persistence diagrams. Substantial efforts have been devoted to this end; among the best-known outcomes are vineyards [12], Fréchet means [24], and persistence landscapes [4].In this work, we provide a new geometric lens with which to view the space of persistence diagrams. Our main result is in fact a statement about the space of (sufficiently tame) persistence modules-these consist of vector spaces and linear maps indexed by the real line R, and their representation theory produces persistence diagrams [23].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Higher Interpolation and Extension for Persistence Modules

Bubenik¹,

Silva²,

Nanda³

2017

SIAM J. Appl. Algebra Geometry

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Abstract. The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of nonexpansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions. Our main construction gives an isometric embedding of a metric space into the metric space of persistence modules with values in the spacetime of this metric space. As a consequence of such "higher interpolation," it becomes possible to compare Vietoris-Rips andČech complexes built within the space of persistence modules. [7,16,22]. The output of this process is a collection of persistent homology groups, which are typically represented via a barcode or a persistence diagram. Recent applications of persistence often confront dynamically evolving data [2,17], and in these cases one requires the ability to make inferences about the dynamics from collections of persistence diagrams. Substantial efforts have been devoted to this end; among the best-known outcomes are vineyards [12], Fréchet means [24], and persistence landscapes [4].In this work, we provide a new geometric lens with which to view the space of persistence diagrams. Our main result is in fact a statement about the space of (sufficiently tame) persistence modules-these consist of vector spaces and linear maps indexed by the real line R, and their representation theory produces persistence diagrams [23]. The class Mod of persistence modules admits an interleaving metric, and the interpolation lemma from [8] establishes that Mod is a path metric space-two modules which are e-interleaved for 0 ≤ e < ∞ can always be connected by a path in Mod of length e. This lemma plays a key role in

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Fréchet Means for Distributions of Persistence Diagrams

Cited by 217 publications

References 26 publications

Topological data analysis of single-trial electroencephalographic signals

Topological data analysis of single-trial electroencephalographic signals

Progressive Wasserstein Barycenters of Persistence Diagrams

Higher Interpolation and Extension for Persistence Modules

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