Abstract

In this paper, we estimate the central mean subspace in a dimension

reduction problem where the response is a symmetric positive-definite matrix.

We propose the intrinsic minimum average variance estimation and the intrinsic

outer product gradient method which fully exploit the geometric structure of the

Riemannian manifold where the response resides. We present algorithms for our

newly developed methods under the log-Euclidean metric and the log-Cholesky

metric.

The two metrics endow the manifold with a commutative Lie group

structure that transforms our manifold model into a Euclidean one and helps us

derive the consistency and asymptotic normality of estimators. Our methods are

then naturally extended to the case allowing p = pn to diverge and the case of

general Riemannian manifolds. Several simulation studies and an application to

the New York taxi network data showcase the superiority of our proposals.

Information

Preprint No.SS-2023-0268
Manuscript IDSS-2023-0268
Complete AuthorsBaiyu Chen, Shuang Dai, Zhou Yu
Corresponding AuthorsZhou Yu
Emailszyu@stat.ecnu.edu.cn

References

  1. Arsigny, V., P. Fillard, X. Pennec, and N. Ayache (2007). Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM Journal on Matrix Analysis and Applications 29(1), 328–347.
  2. Bhattacharjee, S. and H.-G. M¨uller (2023). Single index Fr´echet regression. The Annals of Statistics 51(4), 1770–1798.
  3. Cai, Z., Y. Xia, and W. Hang (2022). An outer-product-of-gradient approach to dimension reduction and its application to classification in high dimensional space. Journal of the American Statistical Association 118(543), 1671–1681.
  4. Cook, R. D. and S. Weisberg (1991). Sliced inverse regression for dimension reduction: Comment. Journal of the American Statistical Association 86(414), 333–335.
  5. Cornea, E., H. Zhu, P. Kim, and J. G. Ibrahim (2017). Regression models on Riemannian symmetric spaces. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 79(2), 463–482.
  6. Dubey, P. and H.-G. M¨uller (2020). Functional models for time-varying random objects. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 82(2), 275–327.
  7. Huettel, S. A., A. W. Song, and G. McCarthy (2008). Functional magnetic resonance imaging. Sinauer Associates.
  8. Kendall, W. S. and H. Le (2021). Limit theorems for empirical fr´echet means of independent and non-identically distributed manifold-valued random variables. Brazilian Journal of Probability and Statistics 25(3), 323–352.
  9. Lang, S. (1999). Fundamentals of Differential Geometry. Springer New York.
  10. Li, B. and S. Wang (2007). On directional regression for dimension reduction. Journal of the American Statistical Association 102(479), 997–1008.
  11. Li, K. C. (1991). Sliced inverse regression for dimension reduction. Journal of the American Statistical Association 86(414), 316–327.
  12. Lin, Z. (2019). Riemannian geometry of symmetric positive definite matrices via Cholesky decomposition. SIAM Journal on Matrix Analysis and Applications 40(4), 1353–1370.
  13. Lin, Z. and H.-G. M¨uller (2021). Total variation regularized fr´echet regression for metric-space valued data. The Annals of Statistics 49(6), 3510–3533.
  14. Lin, Z., H.-G. M¨uller, and B. U. Park (2022). Additive models for symmetric positive-definite matrices and Lie groups. Biometrika 110(2), 361–379.
  15. Lin, Z. and F. Yao (2019). Intrinsic riemannian functional data analysis. The Annals of Statistics 47(6), 3533–3577.
  16. Ma, Y. and L. Zhu (2012). A semiparametric approach to dimension reduction. Journal of the American Statistical Association 107(497), 168–179.
  17. Ma, Y. and L. Zhu (2013). Efficient estimation in sufficient dimension reduction. The Annals of Statistics 41(1), 250–268.
  18. Ma, Y. and L. Zhu (2019). Semiparametric estimation and inference of variance function with large dimensional covariates. Statistica Sinica 29(2), 567–588.
  19. Petersen, A. and H.-G. M¨uller (2019). Fr´echet regression for random objects with Euclidean predictors. The Annals of Statistics 47(2), 691–719.
  20. Sz´ekely, G. J., M. L. Rizzo, and N. K. Bakirov (2007). Measuring and testing dependence by correlation of distances. The Annals of Statistics 35(6), 2769–2794.
  21. Tu, L. W. (2011). An Introduction to Manifolds. Springer New York.
  22. Tucker, D. C., Y. Wu, and H.-G. M¨uller (2021). Variable selection for global Fr´echet regression. Journal of the American Statistical Association 118(542), 1–15.
  23. Xia, Y. (2006). Asymptotic distributions for two estimators of the single-index model. Econometric Theory 22(6), 1112–1137.
  24. Xia, Y. (2007). A constructive approach to the estimation of dimension reduction directions. The Annals of Statistics 35(6), 2654–2690.
  25. Xia, Y., H. Tong, W. K. Li, and L.-X. Zhu (2002). An adaptive estimation of dimension reduction space. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64(3), 363–410.
  26. Ying, C. and Z. Yu (2022). Fr´echet sufficient dimension reduction for random objects. Biometrika 109(4), 975–992.
  27. Yuan, Y., H. Zhu, W. Lin, and J. S. Marron (2012). Local polynomial regression for symmetric positive definite matrices. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 74(4), 697–719.
  28. Zhang, H. (2021). Minimum average variance estimation with group lasso for the multivariate response central mean subspace. Journal of Multivariate Analysis 184, 1–17.
  29. Zhang, Q., L. Xue, and B. Li (2024). Dimension reduction for Fr´echet regression. Journal of the American Statistical Association 119(548), 2733–2747.
  30. Zhou, H., Z. Lin, and F. Yao (2021). Intrinsic Wasserstein correlation analysis. arXiv preprint, arXiv: 2105.15000 [stat.ME].
  31. Zhu, C. and H.-G. M¨uller (2023). Autoregressive optimal transport models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 85(3), 1012–1033.
  32. Zhu, H., Y. Chen, J. G. Ibrahim, Y. Li, C. Hall, and W. Lin (2009). Intrinsic regression models for positive-definite matrices with applications to diffusion tensor imaging. Journal of the American Statistical Association 104(487), 1203–1212.

Acknowledgments

The authors thank the Editor, Associate Editor, and two anonymous reviewers for their constructive feedback on earlier versions of this paper.

The authors contributed equally to this paper and are listed in alphabetical order. The research is supported by the National Key R&D Program of

China (Grant No. 2023YFA1008700 and 2023YFA1008703), the National

Natural Science Foundation of China (Grant No. 12371289), the Basic Research Project of Shanghai Science and Technology Commission (Grant No.

22JC1400800) and the Shanghai Pilot Program for Basic Research (Grant

No. TQ20220105).

Supplementary Materials

Contain: 1) algorithms for iOPG and iMAVE; 2) expressions of asymptotic

covariance matrices in Theorem 2 and 3; 3) convergence results of iOPG

and iMAVE on a general manifold; 4) a simulation study testing the CV

procedure of choosing the structural dimension d and a simulation study

under the general manifold case; 5) details of data collection and processing

in the New York taxi network application; 6) all proofs of theoretical results

that appear in this paper.

Supplementary materials are available for download.