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Sparse-smooth regularized singular value decomposition

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  • Hong, Zhaoping
  • Lian, Heng

Abstract

We consider penalized singular value decomposition (SVD) for a (noisy) data matrix when the left singular vector has a sparse structure and the right singular vector is a discretized function. Such situations typically arise from spatio-temporal data where only some small spatial regions are “activated” as in fMRI data. We use two penalties that impose sparsity and smoothness. However, it is shown, somewhat surprisingly, that the value of only one parameter has to be chosen. This is in stark contrast to the penalized SVD models proposed by Huang et al. (2009) [12] and by Lee et al. (2010) [14]. We carry out some simulation studies and use an artificial fMRI data set and a real data set to illustrate the proposed approach.

Suggested Citation

  • Hong, Zhaoping & Lian, Heng, 2013. "Sparse-smooth regularized singular value decomposition," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 163-174.
    • Handle: RePEc:eee:jmvana:v:117:y:2013:i:c:p:163-174
    DOI: 10.1016/j.jmva.2013.02.011
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    References listed on IDEAS

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    1. Johnstone, Iain M. & Lu, Arthur Yu, 2009. "On Consistency and Sparsity for Principal Components Analysis in High Dimensions," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 682-693.
    2. Díaz-García, José A. & Caro-Lopera, Francisco J., 2012. "Statistical theory of shape under elliptical models and singular value decompositions," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 77-92, January.
    3. Huang, Jianhua Z. & Shen, Haipeng & Buja, Andreas, 2009. "The Analysis of Two-Way Functional Data Using Two-Way Regularized Singular Value Decompositions," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1609-1620.
    4. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
    5. Hu, Jianhua & He, Xuming & Cote, Gilbert J. & Krahe, Ralf, 2009. "Singular Value Decomposition–Based Alternative Splicing Detection," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 944-953.
    6. Reiss, Philip T. & Ogden, R. Todd, 2007. "Functional Principal Component Regression and Functional Partial Least Squares," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 984-996, September.
    7. Mihee Lee & Haipeng Shen & Jianhua Z. Huang & J. S. Marron, 2010. "Biclustering via Sparse Singular Value Decomposition," Biometrics, The International Biometric Society, vol. 66(4), pages 1087-1095, December.
    8. Yata, Kazuyoshi & Aoshima, Makoto, 2010. "Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2060-2077, October.
    9. Antoniadis A. & Fan J., 2001. "Regularization of Wavelet Approximations," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 939-967, September.
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    Keywords

    fMRI; Splines; SVD; Wavelets;
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