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Entropy
Volume 23
Issue 4
10.3390/e23040490
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Article

Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold

by
Clément Mantoux
1,2,3,*,
Baptiste Couvy-Duchesne
1,2,
Federica Cacciamani
1,2,
Stéphane Epelbaum
1,2,4,
Stanley Durrleman
1,2 and
Stéphanie Allassonnière
5,6
1
ARAMIS Project Team, Inria, 75013 Paris, France
2
ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, Hôpital de la Pitié-Salpêtrière, 75013 Paris, France
3
CMAP, École Polytechnique, 91120 Palaiseau, France
4
Institute of Memory and Alzheimer’s Disease (IM2A), Centre of Excellence of Neurodegenerative Disease (CoEN), CIC Neurosciences, AP-HP, Department of Neurology, Hôpital de la Pitié-Salpêtrière, 75013 Paris, France
5
Centre de Recherche des Cordeliers, Université de Paris, INSERM UMR 1138, Sorbonne Université, 75006 Paris, France
6
HEKA Project Team, Inria, 75006 Paris, France
*
Author to whom correspondence should be addressed.
Entropy 2021, 23(4), 490; https://doi.org/10.3390/e23040490
Submission received: 13 March 2021 / Revised: 8 April 2021 / Accepted: 14 April 2021 / Published: 20 April 2021
(This article belongs to the Special Issue Approximate Bayesian Inference)
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Abstract

Network analysis provides a rich framework to model complex phenomena, such as human brain connectivity. It has proven efficient to understand their natural properties and design predictive models. In this paper, we study the variability within groups of networks, i.e., the structure of connection similarities and differences across a set of networks. We propose a statistical framework to model these variations based on manifold-valued latent factors. Each network adjacency matrix is decomposed as a weighted sum of matrix patterns with rank one. Each pattern is described as a random perturbation of a dictionary element. As a hierarchical statistical model, it enables the analysis of heterogeneous populations of adjacency matrices using mixtures. Our framework can also be used to infer the weight of missing edges. We estimate the parameters of the model using an Expectation-Maximization-based algorithm. Experimenting on synthetic data, we show that the algorithm is able to accurately estimate the latent structure in both low and high dimensions. We apply our model on a large data set of functional brain connectivity matrices from the UK Biobank. Our results suggest that the proposed model accurately describes the complex variability in the data set with a small number of degrees of freedom.
Keywords: network modeling; network variability; Stiefel manifold; MCMC-SAEM; data imputation network modeling; network variability; Stiefel manifold; MCMC-SAEM; data imputation

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MDPI and ACS Style

Mantoux, C.; Couvy-Duchesne, B.; Cacciamani, F.; Epelbaum, S.; Durrleman, S.; Allassonnière, S. Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold. Entropy 2021, 23, 490. https://doi.org/10.3390/e23040490

AMA Style

Mantoux C, Couvy-Duchesne B, Cacciamani F, Epelbaum S, Durrleman S, Allassonnière S. Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold. Entropy. 2021; 23(4):490. https://doi.org/10.3390/e23040490

Chicago/Turabian Style

Mantoux, Clément, Baptiste Couvy-Duchesne, Federica Cacciamani, Stéphane Epelbaum, Stanley Durrleman, and Stéphanie Allassonnière. 2021. "Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold" Entropy 23, no. 4: 490. https://doi.org/10.3390/e23040490

APA Style

Mantoux, C., Couvy-Duchesne, B., Cacciamani, F., Epelbaum, S., Durrleman, S., & Allassonnière, S. (2021). Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold. Entropy, 23(4), 490. https://doi.org/10.3390/e23040490

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