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Mantoux C, Durrleman S, Allassonnière S. Asymptotic Analysis of a Matrix Latent Decomposition Model. ESAIM-PROBAB STAT 2022. [DOI: 10.1051/ps/2022004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Abstract
Matrix data sets arise in network analysis for medical applications, where each network belongs to a subject and represents a measurable phenotype. These large dimensional data are often modeled using lower-dimensional latent variables, which explain most of the observed variability and can be used for predictive purposes. In this paper, we provide asymptotic convergence guarantees for the estimation of a hierarchical statistical model for matrix data sets. It captures the variability of matrices by modeling a truncation of their eigendecomposition. We show that this model is identifiable, and that consistent Maximum A Posteriori (MAP) estimation can be performed to estimate the distribution of eigenvalues and eigenvectors. The MAP estimator is shown to be asymptotically normal for a restricted version of the model.
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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:e23040490. [PMID: 33924060 PMCID: PMC8074266 DOI: 10.3390/e23040490] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/13/2021] [Revised: 04/08/2021] [Accepted: 04/14/2021] [Indexed: 11/22/2022]
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.
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Affiliation(s)
- Clément Mantoux
- ARAMIS Project Team, Inria, 75013 Paris, France; (B.-C.D.); (F.C.); (S.E.); (S.D.)
- ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, Hôpital de la Pitié-Salpêtrière, 75013 Paris, France
- CMAP, École Polytechnique, 91120 Palaiseau, France
- Correspondence:
| | - Baptiste Couvy-Duchesne
- ARAMIS Project Team, Inria, 75013 Paris, France; (B.-C.D.); (F.C.); (S.E.); (S.D.)
- ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, Hôpital de la Pitié-Salpêtrière, 75013 Paris, France
| | - Federica Cacciamani
- ARAMIS Project Team, Inria, 75013 Paris, France; (B.-C.D.); (F.C.); (S.E.); (S.D.)
- ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, Hôpital de la Pitié-Salpêtrière, 75013 Paris, France
| | - Stéphane Epelbaum
- ARAMIS Project Team, Inria, 75013 Paris, France; (B.-C.D.); (F.C.); (S.E.); (S.D.)
- ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, Hôpital de la Pitié-Salpêtrière, 75013 Paris, France
- 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
| | - Stanley Durrleman
- ARAMIS Project Team, Inria, 75013 Paris, France; (B.-C.D.); (F.C.); (S.E.); (S.D.)
- ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, Hôpital de la Pitié-Salpêtrière, 75013 Paris, France
| | - Stéphanie Allassonnière
- Centre de Recherche des Cordeliers, Université de Paris, INSERM UMR 1138, Sorbonne Université, 75006 Paris, France;
- HEKA Project Team, Inria, 75006 Paris, France
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