The contractivity of cone-preserving multilinear mappings

Gautier, Antoine; Tudisco, Francesco

doi:10.1088/1361-6544/ab3352

Cited by 13 publications

(12 citation statements)

References 37 publications

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“…Here, we quantified the manifold structure of CCPs (embedding dataset) by computing a core-quality value [33] for each node (connection) of the embedding dataset V . Core-periphery detection was performed by using a nonlinear spectral (NSM) algorithm [33] derived from work on the nonlinear Perron-Frobenius theory [34,35,36]. This approach assigns a nonnegative value to each node such that a smaller value indicates a lower level of importance of that node in the manifold structure.…”

Section: Methodsmentioning

confidence: 99%

Manifold Learning of Dynamic Functional Connectivity Reliably Identifies Functionally Consistent Coupling Patterns in Human Brains

Yang

Wang

et al. 2019

Brain Sciences

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Most previous work on dynamic functional connectivity (dFC) has focused on analyzing temporal traits of functional connectivity (similar coupling patterns at different timepoints), dividing them into functional connectivity states and detecting their between-group differences. However, the coherent functional connectivity of brain activity among the temporal dynamics of functional connectivity remains unknown. In the study, we applied manifold learning of local linear embedding to explore the consistent coupling patterns (CCPs) that reflect functionally homogeneous regions underlying dFC throughout the entire scanning period. By embedding the whole-brain functional connectivity in a low-dimensional manifold space based on the Human Connectome Project (HCP) resting-state data, we identified ten stable patterns of functional coupling across regions that underpin the temporal evolution of dFC. Moreover, some of these CCPs exhibited significant neurophysiological meaning. Furthermore, we apply this method to HCP rsfMR and tfMRI data as well as sleep-deprivation data and found that the topological organization of these low-dimensional structures has high potential for predicting sleep-deprivation states (classification accuracy of 92.3%) and task types (100% identification for all seven tasks).In summary, this work provides a methodology for distilling coherent low-dimensional functional connectivity structures in complex brain dynamics that play an important role in performing tasks or characterizing specific states of the brain.

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Section: Methodsmentioning

confidence: 99%

Manifold Learning of Dynamic Functional Connectivity Reliably Identifies Functionally Consistent Coupling Patterns in Human Brains

Yang

Wang

et al. 2019

Brain Sciences

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“…Tensors can also represent higher-order data and are broadly used in machine learning [5,27,43,47,53]. We analyze the iterations of NHOLS as a type of tensor contraction; from this, we extend recent nonlinear Perron-Frobenius theory [19,20] to establish convergence results.…”

Section: Related Workmentioning

confidence: 99%

Nonlinear Higher-Order Label Spreading

Tudisco

Benson

Prokopchik

2021

Proceedings of the Web Conference 2021

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Label spreading is a general technique for semi-supervised learning with point cloud or network data, which can be interpreted as a diffusion of labels on a graph. While there are many variants of label spreading, nearly all of them are linear models, where the incoming information to a node is a weighted sum of information from neighboring nodes. Here, we add nonlinearity to label spreading through nonlinear functions of higher-order structure in the graph, namely triangles in the graph. For a broad class of nonlinear functions, we prove convergence of our nonlinear higher-order label spreading algorithm to the global solution of a constrained semi-supervised loss function. We demonstrate the efficiency and efficacy of our approach on a variety of point cloud and network datasets, where the nonlinear higher-order model compares favorably to classical label spreading, as well as hypergraph models and graph neural networks.

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“…k = 2). However, it has been observed in, e.g., [18,32,20] that nonlinear eigenvector equations of the form (4) admit a unique solution that can be computed to an arbitrary precision if the tensor A is not too sparse and if the exponent p satisfies certain assumptions.…”

Section: Tensor-based Eigenvector Centrality For Uniform Hypergraphsmentioning

confidence: 99%

Node and Edge Eigenvector Centrality for Hypergraphs

Tudisco

Higham

2021

Preprint

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Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron-Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.

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The contractivity of cone-preserving multilinear mappings

Cited by 13 publications

References 37 publications

Manifold Learning of Dynamic Functional Connectivity Reliably Identifies Functionally Consistent Coupling Patterns in Human Brains

Manifold Learning of Dynamic Functional Connectivity Reliably Identifies Functionally Consistent Coupling Patterns in Human Brains

Nonlinear Higher-Order Label Spreading

Node and Edge Eigenvector Centrality for Hypergraphs

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