Abstract
Exploring the relationships between variables of data has attracted a lot of attentions in the fields like machine learning, signal processing, neuroscience, and social network. It enables us to better understand the data before choosing models for particular tasks. Gaussian graphical model provides a natural and straightforward way to reveal the relationships of variables in data, where the zero entries in precision matrices represent conditional independence between corresponding variables and vice versa. However, traditional vector- or matrix-variate-based methods are not suitable for higher-order data analysis due to the possible structural information loss, and this is where tensor-variate-based Gaussian graphical model comes in. In this chapter, we discuss the development of Gaussian graphical models from vector-variate-based methods, matrix-variate-based methods to tensor-variate-based methods and mainly focus on the last one. Detailed assumptions, probability density functions, optimization models based on maximum likelihood estimation, and some typical algorithms are given. In addition, we illustrate the applications of tensor Gaussian graphical models in environmental prediction and mice aging study and give some numerically experimental results.
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References
Banerjee, O., El Ghaoui, L., d’Aspremont, A.: Model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data. J. Mach. Learn. Res. 9, 485–516 (2008)
Beal, M.J., Jojic, N., Attias, H.: A graphical model for audiovisual object tracking. IEEE Trans. Pattern Anal. Mach. Intell. 25(7), 828–836 (2003)
Brouard, C., de Givry, S., Schiex, T.: Pushing data into CP models using graphical model learning and solving. In: International Conference on Principles and Practice of Constraint Programming, pp. 811–827. Springer, Berlin (2020)
d’Aspremont, A., Banerjee, O., El Ghaoui, L.: First-order methods for sparse covariance selection. SIAM J. Matrix Anal. Appl. 30(1), 56–66 (2008)
Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348–1360 (2001)
Friedman, J., Hastie, T., Tibshirani, R.: Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3), 432–441 (2008)
He, S., Yin, J., Li, H., Wang, X.: Graphical model selection and estimation for high dimensional tensor data. J. Multivar. Anal. 128, 165–185 (2014)
Leng, C., Tang, C.Y.: Sparse matrix graphical models. J. Am. Stat. Assoc. 107(499), 1187–1200 (2012)
Lyu, X., Sun, W.W., Wang, Z., Liu, H., Yang, J., Cheng, G.: Tensor graphical model: non-convex optimization and statistical inference. IEEE Trans. Pattern Anal. Mach. Intell. 42(8), 2024–2037 (2019)
Ma, C., Lu, J., Liu, H.: Inter-subject analysis: a partial Gaussian graphical model approach. J. Am. Stat. Assoc. 116(534), 746–755 (2021)
Meinshausen, N., Bühlmann, P., et al.: High-dimensional graphs and variable selection with the lasso. Ann. Stat. 34(3), 1436–1462 (2006)
Shahid, N., Grassi, F., Vandergheynst, P.: Multilinear low-rank tensors on graphs & applications (2016, preprint). arXiv:1611.04835
Shen, X., Pan, W., Zhu, Y.: Likelihood-based selection and sharp parameter estimation. J. Am. Stat. Assoc. 107(497), 223–232 (2012)
Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. B 58(1), 267–288 (1996)
Wu, W.B., Pourahmadi, M.: Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika 90(4), 831–844 (2003)
Xu, P., Zhang, T., Gu, Q.: Efficient algorithm for sparse tensor-variate gaussian graphical models via gradient descent. In: Artificial Intelligence and Statistics, pp. 923–932. PMLR, Westminster (2017)
Yin, J., Li, H.: Model selection and estimation in the matrix normal graphical model. J. Multivariate Anal. 107, 119–140 (2012)
Yuan, M.: High dimensional inverse covariance matrix estimation via linear programming. J. Mach. Learn. Res. 11, 2261–2286 (2010)
Yuan, M., Lin, Y.: Model selection and estimation in the gaussian graphical model. Biometrika 94(1), 19–35 (2007)
Zhang, C.H., et al.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38(2), 894–942 (2010)
Zhou, S., et al.: Gemini: graph estimation with matrix variate normal instances. Ann. Stat. 42(2), 532–562 (2014)
Zou, H.: The adaptive lasso and its oracle properties. J. Am. Stat. Assoc. 101(476), 1418–1429 (2006)
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Liu, Y., Liu, J., Long, Z., Zhu, C. (2022). Tensor-Based Gaussian Graphical Model. In: Tensor Computation for Data Analysis. Springer, Cham. https://doi.org/10.1007/978-3-030-74386-4_12
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DOI: https://doi.org/10.1007/978-3-030-74386-4_12
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