Variational Wasserstein Clustering

Liang, Mao; Zhang, Wen; Gu, Xianfeng; Wang, Yalin

doi:10.1007/978-3-030-01267-0_20

Cited by 31 publications

(22 citation statements)

References 33 publications

(56 reference statements)

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“…Effective use of such topological descriptors requires a notion of proximity that quantifies the similarity between persistence barcodes, a convenient representation for connected components and cycles [Ghrist, 2008]. Wasserstein distance, which measures the minimal effort to modify one persistence barcode to another [Rabin et al, 2011], is an excellent choice due to its appealing geometric properties [Staerman et al, 2021] and its effectiveness shown in many machine learning applications [Kolouri et al, 2017, Mi et al, 2018, Solomon et al, 2015. Importantly, Wasserstein distance can be used to interpolate networks while preserving topological structure [Songdechakraiwut et al, 2021], and the mean under the Wasserstein distance, known as Wasserstein barycenter [Agueh and Carlier, 2011], can be viewed as the topological centroid of a set of networks.…”

Section: Introductionmentioning

confidence: 99%

“…The high cost of computing persistence barcodes, Wasserstein distance and the Wasserstein barycenter limit their applications to small scale problems, see, e.g., [Clough et al, 2020, Hu et al, 2019, Kolouri et al, 2017, Mi et al, 2018. Although approximation algorithms have been developed [Cuturi, 2013, Cuturi and Doucet, 2014, Lacombe et al, 2018, Li et al, 2020, Solomon et al, 2015, Vidal et al, 2019, Xie et al, 2020, Ye et al, 2017, it is unclear whether these approximations are effective for clustering complex networks as they inevitably limit sensitivity to subtle topological features.…”

Section: Introductionmentioning

confidence: 99%

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Fast Topological Clustering with Wasserstein Distance

Songdechakraiwut¹,

Krause²,

Banks³

et al. 2021

Preprint

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The topological patterns exhibited by many real-world networks motivate the development of topology-based methods for assessing the similarity of networks. However, extracting topological structure is difficult, especially for large and dense networks whose node degrees range over multiple orders of magnitude. In this paper, we propose a novel and computationally practical topological clustering method that clusters complex networks with intricate topology using principled theory from persistent homology and optimal transport. Such networks are aggregated into clusters through a centroid-based clustering strategy based on both their topological and geometric structure, preserving correspondence between nodes in different networks. The notions of topological proximity and centroid are characterized using a novel and efficient approach to computation of the Wasserstein distance and barycenter for persistence barcodes associated with connected components and cycles. The proposed method is demonstrated to be effective using both simulated networks and measured functional brain networks.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fast Topological Clustering with Wasserstein Distance

Songdechakraiwut¹,

Krause²,

Banks³

et al. 2021

Preprint

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“…An approach to empirical distribution clustering via k-means is also given in Henderson, Gallagher and Eliassi-Rad [HGER15] in a non-financial context. Other works have utilised the Wasserstein distance for clustering problems, see for instance [LW08,YWWL17], where in the latter distributions are represented as weight-mass pairs, and clustering is considered in the context of images and documents, or [MZGW18] for an approach using variational optimal transport. Such approaches are similar to the work in this paper as they often employ classic unsupervised learning algorithms with some modification that allows them to handle distributional datum.…”

Section: Introductionmentioning

confidence: 99%

Clustering Market Regimes using the Wasserstein Distance

Horvath¹,

Issa²,

Muguruza³

2021

Preprint

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The problem of rapid and automated detection of distinct market regimes is a topic of great interest to financial mathematicians and practitioners alike. In this paper, we outline an unsupervised learning algorithm for clustering financial time-series into a suitable number of temporal segments (market regimes). As a special case of the above, we develop a robust algorithm that automates the process of classifying market regimes. The method is robust in the sense that it does not depend on modelling assumptions of the underlying time series as our experiments with real datasets show. This method -dubbed the Wasserstein k-means algorithm -frames such a problem as one on the space of probability measures with finite p th moment, in terms of the p-Wasserstein distance between (empirical) distributions. We compare our WK-means approach with a more traditional clustering algorithms by studying the so-called maximum mean discrepancy scores between, and within clusters. In both cases it is shown that the WK-means algorithm vastly outperforms all considered competitor approaches. We demonstrate the performance of all approaches both in a controlled environment on synthetic data, and on real data.

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“…In this paper, by generalizing our prior work on volumetric Wasserstein distance computation (Mi et al 2017(Mi et al , 2018, we propose a framework to compute the volumetric Wasserstein distance of structural MR images and explore its application as a potential univariate neurodegenerative biomarker. With the proposed framework, a volumetric Wasserstein distance will be computed for each MR image from its optimal transportation (OT) map to the template image.…”

Section: Introductionmentioning

confidence: 99%

Computing Univariate Neurodegenerative Biomarkers with Volumetric Optimal Transportation: A Pilot Study

Liang

Zhang

et al. 2020

Neuroinform

Self Cite

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Changes in cognitive performance due to neurodegenerative diseases such as Alzheimer's disease (AD) are closely correlated to the brain structure alteration. A univariate and personalized neurodegenerative biomarker with strong statistical power based on magnetic resonance imaging (MRI) will benefit clinical diagnosis and prognosis of neurodegenerative diseases. However, few biomarkers of this type have been developed, especially those that are robust to image noise and applicable to clinical analyses. In this paper, we introduce a variational framework to compute optimal transportation (OT) on brain structural MRI volumes and develop a univariate neuroimaging index based on OT to quantify neurodegenerative alterations. Specifically, we compute the OT from each image to a template and measure the Wasserstein distance between them. The obtained Wasserstein distance, Wasserstein Index (WI) for short to specify the distance to a template, is concise, informative and robust to random noise. Comparing to the popular linear programming-based OT computation method, our framework makes use of Newton's method, ǂ Data used in the preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database

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Variational Wasserstein Clustering

Cited by 31 publications

References 33 publications

Fast Topological Clustering with Wasserstein Distance

Fast Topological Clustering with Wasserstein Distance

Clustering Market Regimes using the Wasserstein Distance

Computing Univariate Neurodegenerative Biomarkers with Volumetric Optimal Transportation: A Pilot Study

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