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.
Keywords: Hierarchical model, matrix data sets, low rank, stiefel manifold, identifiability, strong consistency, asymptotic normality
@article{PS_2022__26_1_208_0,
author = {Mantoux, Cl\'ement and Durrleman, Stanley and Allassonni\`ere, St\'ephanie},
title = {Asymptotic {Analysis} of a {Matrix} {Latent} {Decomposition} {Model}},
journal = {ESAIM: Probability and Statistics},
pages = {208--242},
year = {2022},
publisher = {EDP-Sciences},
volume = {26},
doi = {10.1051/ps/2022004},
mrnumber = {4425001},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2022004/}
}
TY - JOUR AU - Mantoux, Clément AU - Durrleman, Stanley AU - Allassonnière, Stéphanie TI - Asymptotic Analysis of a Matrix Latent Decomposition Model JO - ESAIM: Probability and Statistics PY - 2022 SP - 208 EP - 242 VL - 26 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2022004/ DO - 10.1051/ps/2022004 LA - en ID - PS_2022__26_1_208_0 ER -
%0 Journal Article %A Mantoux, Clément %A Durrleman, Stanley %A Allassonnière, Stéphanie %T Asymptotic Analysis of a Matrix Latent Decomposition Model %J ESAIM: Probability and Statistics %D 2022 %P 208-242 %V 26 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ps/2022004/ %R 10.1051/ps/2022004 %G en %F PS_2022__26_1_208_0
Mantoux, Clément; Durrleman, Stanley; Allassonnière, Stéphanie. Asymptotic Analysis of a Matrix Latent Decomposition Model. ESAIM: Probability and Statistics, Tome 26 (2022), pp. 208-242. doi: 10.1051/ps/2022004
[1] , and , Learning Latent Block Structure in Weighted Networks. J. Complex Netw. 3 (2015) 221–248. | MR | DOI
[2] and , Classification of matrix-variate fisher—bingham distribution via maximum likelihood estimation using manifold valued data. Neurocomputing 295 (2018) 72–85. | DOI
[3] , and , Toward a coherent statistical framework for dense deformable template estimation. J. Royal Stat. Soc. B 69 (2007) 3–29. | MR | DOI
[4] , and , Identifiability of parameters in latent structure models with many observed variables. Ann. Stat. 37 (2009) 3099–3132. | MR | Zbl | DOI
[5] and , The asymptotic normal distribution of estimators in factor analysis under general conditions. Ann. Stat. 16 (1988) 759–771. | MR | Zbl | DOI
[6] -Nielsen, Identifiability of mixtures of exponential families. J. Math. Anal. Appl. 12 (1965) 115–121. | MR | Zbl | DOI
[7] , Information and exponential families. In: Statistical theory, Wiley series in probability and mathematical statistics. Wiley, Chichester, New York (1978). | MR | Zbl
[8] , and , Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Stat. 26 (1998) 1614–1635. | MR | Zbl | DOI
[9] and , Consistent noisy independent component analysis. J. Econ. 149 (2009) 12–25. | MR | DOI
[10] , , , , , and , Estimating common harmonic waves of brain networks on Stiefel manifold, in , , , , , , and (editors), Medical Image Computing and Computer Assisted Intervention — MICCAI 2020, Lecture Notes in Computer Science, Springer International Publishing, Cham (2020) 367–367.
[11] , and , A coherent framework for learning spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data. SIAM J. Imag. Sci. 14 (2021) 349–388. | MR | DOI
[12] , Concentrated matrix Langevin distributions. J. Multivar. Anal. 85 (2003) 375–394. | MR | Zbl | DOI
[13] , Statistics on Special Manifolds, Lecture Notes in Statistics, Springer-Verlag, New York (2003). | MR | Zbl
[14] , State space models on special manifolds. J. Multivar. Anal. 97 (2006) 1284–1294. | MR | Zbl | DOI
[15] , Non Singularity of the Asymptotic Fisher Information Matrix in Hidden Markov Models. (2005). | arXiv
[16] , , and , Consistency of the maximum likelihood estimator for general hidden Markov models. Ann. Stat. 39 (2011) 474–513. | MR | Zbl | DOI
[17] , and , Necessary and sufficient conditions for the identifiability of observation-driven models. J. Time Ser. Anal. 42 (2021) 140–160. | MR | DOI
[18] , , , and , A generative-discriminative basis learning framework to predict clinical severity from resting state functional MRI data, in , , , and (editors), Medical Image Computing and Computer Assisted Intervention — MICCAI 2018. Springer International Publishing, Cham (2018), vol. 11072, 163–163. | DOI
[19] , , , and , Integrating neural networks and dictionary learning for multidimensional clinical characterizations from functional connectomics data, in , , , , , , and (editors), Medical Image Computing and Computer Assisted Intervention — MICCAI 2019. Springer International Publishing, Cham (2019), vol. 11766, 709–709. | DOI
[20] , and , Spiked Laplacian Graphs: Bayesian Community Detection in Heterogeneous Networks. [stat] (2020). | arXiv
[21] , and , The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 (1998) 303–353. | MR | Zbl | DOI
[22] , On a theorem of weyl concerning eigenvalues of linear transformations I. Proc. Natl. Acad. Sci. 35 (1949) 652–655. | MR | Zbl | DOI
[23] , Log-gases and random matrices (LMS-34). Vol. 34 of London Mathematical Society Monographs. Princeton University Press (2010). | MR | Zbl
[24] , and , Optimization over the Stiefel Manifold, in vol. 7 of PAMM: Proceedings in Applied Mathematics and Mechanics. Wiley Online Library (2007) 1062205–1062205. | DOI
[25] and , Identifiability of Hierarchical Latent Attribute Models. [cs, stat] (2021). | arXiv | MR
[26] , Simulation of the matrix Bingham—von Mises—Fisher distribution, with applications to multivariate and relational data. J. Comput. Graph. Stat. 18 (2009) 438–456. | MR | DOI
[27] , and , Identifiability of finite mixtures - with applications to circular distributions. Sankhya 66 (2004) 440–449. | MR | Zbl
[28] , Graphons, Cut Norm and Distance, Couplings and Rearrangements, Vol. 4 of New York Journal of Mathematics. NYJM Monographs, State University of New York, University at Albany, Albany, NY 4 (2013) 76. | MR | Zbl
[29] , and , Random Orthogonal Matrices and the Cayley Transform. Bernoulli 26 (2020) 1560–1586. | MR | DOI
[30] and , Maximum Likelihood Estimators for the Matrix Von Mises-Fisher and Bingham Distributions. Ann. Stat. 7 (1979) 599–606. | MR | Zbl
[31] , Identifiability of Finite Mixtures for Directional Data, Ann. Stat. 11 (1983). | MR | Zbl
[32] and , The von Mises—Fisher Matrix Distribution in Orientation Statistics. J.R. Stat Soc. Ser. B (Methodological) 39 (1977) 95–106. | MR | Zbl | DOI
[33] and , Cheeger Inequalities for Graph Limits, [math] (2018). | arXiv
[34] and , Semi-Supervised Classification with Graph Convolutional Networks, in ICLR 2017 (2017).
[35] , and , Saddlepoint Approximations for the Normalizing Constant of Fisher—Bingham Distributions on Products of Spheres and Stiefel Manifolds. Biometrika 100 (2013) 971–984. | MR | Zbl | DOI
[36] and , Variational Bayes Model Averaging for Graphon Functions and Motif Frequencies Inference in -graph Models, Stat. Comput. 26 (2016) 1173–1173. | MR | DOI
[37] and , What Do We Mean by Identifiability in Mixed Effects Models?. J. Pharmacokinet. Pharmacodyn. 43 (2016) 111–122. | DOI
[38] and , Theory of Point Estimation, Springer Texts in Statistics, 2nd edn., Springer, New York (2003). | MR | Zbl
[39] , , , , and , Graph Neural Network for Interpreting Task-fMRI Biomarkers, in , , , , , , and (editors), Medical Image Computing and Computer Assisted Intervention — MICCAI 2019, Lecture Notes in Computer Science, Springer International Publishing, Cham (2019) 485–485.
[40] , , and , On Generalizing Trace Minimization. [cs, math] (2021). | arXiv | MR
[41] , and , Bayesian Nonparametric Inference on the Stiefel Manifold. Stat. Sin. 27 (2017) 535–553. | MR
[42] , Large Networks and Graph Limits. Colloquium Publications, vol. 60, American Mathematical Society, Providence, Rhode Island (2012). | MR | Zbl
[43] , , , , and , Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold, Entropy 23 (2021) 490. | MR | DOI
[44] and , Graphon Estimation from Partially Observed Network Data. [cs, stat] (2019). | arXiv
[45] and , Network Histograms and Universality of Blockmodel Approximation. Proc. Natl. Acad. Sci. 111 (2014) 14722–14727. | DOI
[46] , , and , Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold. Bayesian Anal. 15 (2020) 871–908. | MR
[47] , Bayesian Stochastic Blockmodeling, in , and (editors), Advances in Network Clustering and Blockmodeling, Wiley Series in Computational and Quantitative Social Science, 289–332, Wiley (2020) .
[48] , , and , Asymptotic Normality and Optimalities in Estimation of Large Gaussian Graphical Models. Ann. Stat. 43 (2015). | MR
[49] and , Reconstruction of a Low-Rank Matrix in the Presence of Gaussian Noise. J. Multivar. Anal. 118 (2013) 67–76. | MR | Zbl | DOI
[50] and , EM-based Smooth Graphon Estimation Using MCMC and Spline-Based Approaches. Soc. Netw. 68 (2022) 279–295. | DOI
[51] , and , A Note on the Identifiability of Latent Variable Models for Mixed Longitudinal Data. Stat. Probab. Lett. 167 (2020) 108882. | MR | Zbl | DOI
[52] , Identifiability of Finite Mixtures. Ann. Math. Stat. 34 (1963) 1265–1269. | MR | Zbl | DOI
[53] , Change of Variables for Hausdorff Measure (from the Beginning). Universitá degli Studi di Trieste. Dipartimento di Scienze Matematiche 26 suppl. (1994) 327–327. | MR | Zbl
[54] , Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics, 1st edn., Cambridge Univ. Press, Cambridge (1998). | MR | Zbl
[55] , Rates of Convergence of Spectral Methods for Graphon Estimation, in International Conference on Machine Learning (2018) 5433–5433.
[56] and , On the Identifiability of Finite Mixtures. Ann. Math. Stat. 39 (1968) 209–214. | MR | Zbl | DOI
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