Hypergraph Dissimilarity Measures

Surana, Amit; Chen, Can; Rajapakse, Indika

doi:10.48550/arxiv.2106.08206

Cited by 3 publications

(7 citation statements)

References 44 publications

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“…Our framework thus enables study of explicit structure-function relationships that are observed directly from data, without needing to infer multi-way contacts. In engineering and social systems, hypergraph representation of data has revealed higher-order organization principles efficiently [15,16,17,44]. Our work here extends the application of hypergraphs, demonstrating a natural way to represent and analyze genome organization across scales.…”

Section: Discussionmentioning

confidence: 75%

“…Our framework thus enables study of explicit structure-function relationships that are observed directly from data, eliminating the need for inference of multi-way contacts. The increased precision of hypergraph representation has the potential to reveal patterns of higher-order differential chromatin organization between multiple cell-types, and presents the exciting possibility of application at the single cell-level [11, 12, 13, 40]. Long-range, inter-chromosomal interactions offer a great deal of organization and structure to the genome.…”

Section: Discussionmentioning

confidence: 99%

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Deciphering Multi-way Interactions in the Human Genome

Lindsly

Chen

Dilworth³

et al. 2021

Preprint

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Chromatin architecture, a key regulator of gene expression, is inferred through chromatin contacts. However, classical analyses of chromosome conformation data do not preserve multi-way relationships. Here we use long sequencing reads to map genome-wide multi-way contacts and investigate higher order chromatin organization of the human genome. We use the theory of hypergraphs for data representation and analysis, and quantify higher order structures in primary human fibroblasts and B lymphocytes. Through integration of multi-way contact data with chromatin accessibility, gene expression, and transcription factor binding data, we introduce a data-driven method to extract transcriptional clusters.

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Section: Discussionmentioning

confidence: 75%

Section: Discussionmentioning

confidence: 99%

Deciphering Multi-way Interactions in the Human Genome

Lindsly

Chen

Dilworth³

et al. 2021

Preprint

Self Cite

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“…Another relevant topic is the study of hypergraph similarity or dissimilarity measures (e.g., [40]). The first type of approach [40] transforms a hypergraph into a graph and applies standard graph similarity or dissimilarity measures.…”

Section: Related Workmentioning

confidence: 99%

“…As tensors [22] are becoming increasingly popular in modeling higher-order interactions, the second type of approach considers a hypergraph as a tensor [2], and study the distance between a pair of tensor representations [40]. The work in [16] describes a principled framework for hypergraph matching and proposed to use this framework for going beyond isometry-invariant shape matching and obtaining invariance under similarity, affine, and projective transformations.…”

Section: Related Workmentioning

confidence: 99%

Hypergraph Co-Optimal Transport: Metric and Categorical Properties

Chowdhury¹,

Needham²,

Semrad³

et al. 2021

Preprint

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Hypergraphs capture multi-way relationships in data, and they have consequently seen a number of applications in higher-order network analysis, computer vision, geometry processing, and machine learning. In this paper, we develop the theoretical foundations in studying the space of hypergraphs using ingredients from optimal transport. By enriching a hypergraph with probability measures on its nodes and hyperedges, as well as relational information capturing local and global structure, we obtain a general and robust framework for studying the collection of all hypergraphs. First, we introduce a hypergraph distance based on the co-optimal transport framework of Redko et al. and study its theoretical properties. Second, we formalize common methods for transforming a hypergraph into a graph as maps from the space of hypergraphs to the space of graphs and study their functorial properties and Lipschitz bounds. Finally, we demonstrate the versatility of our Hypergraph Co-Optimal Transport (HyperCOT) framework through various examples.Recent years have seen the extension of the Gromov-Wasserstein (GW) framework-originally developed as a tool for comparing metric measure spaces [27, 28]-to probabilistic graph matching tasks [36,20,46,45,9,43,10]. The numerous benefits of this approach include computability via gradient descent [31,17] or backpropagation [44], state-of-the-art performance in tasks such as graph partitioning [45,12], and an underlying theoretical Riemannian framework [38,11]. These successes motivate the development of a GW framework for hypergraphs, which is the goal of this paper. Our contributions include:

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“…There are many different notions of tensor eigenvalues such as Heigenvalues, Z-eigenvalues, M-eigenvalues, and U-eigenvalues [10,29,30], which have different applications in network theory, machine learning, elasticity theory, and dynamical systems. Surana et al [38] compared the H-eigenvalue spectrum between the two Laplacian tensors for measuring hypergraph distance. Chen et al [13] showed that the Z-eigenvector associated with the second smallest Z-eigenvalue of a normalized Laplacian tensor can be used for hypergraph partition.…”

Section: Introductionmentioning

confidence: 99%

Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems

Chen¹

2021

Preprint

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In this paper, we investigate the explicit solutions and stability properties of certain continuous-time homogeneous polynomial dynamical systems via tensor algebra. In particular, if a system of homogeneous polynomial differential equations can be represented by an orthogonal decomposable tensor, we can write down its explicit solution in a simple fashion by exploiting tensor Z-eigenvalues and Z-eigenvectors. In addition, according to the form of the explicit solution, we explore the stability properties of the homogeneous polynomial dynamical system. We discover that the Z-eigenvalues from the orthogonal decomposition of the corresponding dynamic tensor can offer necessary and sufficient stability conditions, similar to these from linear systems theory. Finally, we demonstrate our results with numerical examples.

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Hypergraph Dissimilarity Measures

Cited by 3 publications

References 44 publications

Deciphering Multi-way Interactions in the Human Genome

Deciphering Multi-way Interactions in the Human Genome

Hypergraph Co-Optimal Transport: Metric and Categorical Properties

Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems

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