Abstract

Image data exhibit local correlations and high-dimensional structure,

with their covariance matrix capturing spatial dependencies and governing performance in feature extraction, dimensionality reduction, and reconstruction.

Two-dimensional principal component analysis (2D-PCA) effectively preserves

row/column-wise correlations, enabling accurate covariance estimation and identification of meaningful components. However, the limiting spectral behavior and

phase transition phenomena of image covariance matrices in high-dimensional

settings remain theoretically unexplored, compromising inference tasks such as

principal component determination. To bridge this gap, we establish a unified

theoretical framework for the asymptotic spectral behavior of the image covariance matrix under general non-i.i.d. assumptions, systematically characterizing

its phase transition from non-spiked to spiked regimes and deriving corresponding central limit theorems for spectral statistics.

Building on this theoretical

foundation, we propose a rigorous sequential hypothesis testing procedure to determine the number of principal components, offering provably consistent and

prior-free detection thresholds that ensure statistical power in high dimensions.

Our method is validated through extensive simulations and on real-world facial

image datasets, demonstrating superior accuracy and robustness over existing

approaches. The resulting framework provides a plug-and-play spectral module

for diverse image processing and machine learning applications.

Key words and phrases: Random Matrix Theory, 2D-PCA, Image Covariance Matrix

Information

Preprint No.SS-2026-0076
Manuscript IDSS-2026-0076
Complete AuthorsShuhang Liu, Li Yang, Dandan Jiang
Corresponding AuthorsDandan Jiang
Emailsjiangdd@xjtu.edu.cn

References

  1. Ades-Aron, B., G. Lemberskiy, J. Veraart, and J. Golfinos (2021). Improved task-based functional MRI language mapping in patients with brain tumors through Marchenko–Pastur principal component analysis denoising. Radiology 298(2), 365–373.
  2. Bai, Z. and J. W. Silverstein (2008). CLT for linear spectral statistics of large-dimensional sample covariance matrices. In Advances in Statistics, Singapore, pp. 281–333. World Scientific.
  3. Bai, Z. and J. Yao (2012). On sample eigenvalues in a generalized spiked population model. Journal of Multivariate Analysis 106, 167–177.
  4. Bai, Z. D., B. Miao, and G. Pan (2007). On asymptotics of eigenvectors of large sample covariance matrix. The Annals of Probability 35(4), 1532–1572.
  5. Bao, Z., X. Ding, J. Wang, and K. Wang (2022). Statistical inference for principal components of spiked covariance matrices. The Annals of Statistics 50(2), 1144–1169.
  6. Chang, Y., L. Yan, and S. Zhong (2017). Hyper-laplacian regularized unidirectional low-rank tensor recovery for multispectral image denoising. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4260–4268.
  7. Ding, K., K. Ma, S. Wang, and E. P. Simoncelli (2022). Image quality assessment: Unifying structure and texture similarity. IEEE Transactions on Pattern Analysis and Machine Intelligence 44(5), 2567–2581.
  8. Ding, X. and F. Yang (2021). Spiked separable covariance matrices and principal components. The Annals of Statistics 49(2), 1113–1138.
  9. Georghiades, A. S., P. N. Belhumeur, and D. J. Kriegman (2001). From few to many: Illumination cone models for face recognition under variable lighting and pose. IEEE Transactions on Pattern Analysis and Machine Intelligence 23(6), 643–660.
  10. Guo, S., Z. Yan, K. Zhang, W. Zuo, and L. Zhang (2019). Toward convolutional blind denoising of real photographs. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1712–1722.
  11. Jiang, D. (2023). A universal test on spikes in a high-dimensional generalized spiked model and its applications. Statistica Sinica 33, 1749–1770.
  12. Jiang, D. and Z. Bai (2021). Generalized four moment theorem and an application to CLT for spiked eigenvalues of high-dimensional covariance matrices. Bernoulli 27(1), 274–294.
  13. Jin, S., K. Miao, and L. Su (2021). On factor models with random missing: EM estimation, inference, and cross validation. Journal of Econometrics 222(1), 745–777.
  14. Ledoit, O. and M. Wolf (2020). Analytical nonlinear shrinkage of large-dimensional covariance matrices. The Annals of Statistics 48(5), 3043–3065.
  15. Li, Z. and Y. Kuang (2024). SPA: Signflip parallel analysis to optimize the number of principal components in two-dimensional PCA. Journal of Data Science.
  16. Minka, T. P. (2000). Automatic choice of dimensionality for PCA. In Advances in Neural Information Processing Systems, Volume 13, Cambridge, MA, pp. 598–604. MIT Press.
  17. Wang, M., X. Jiang, J. Gao, T. Wang, C. Hu, and F. Liu (2021). Minimum unbiased risk estimate based 2DPCA for color image denoising. Neurocomputing 440, 127–144.
  18. Xu, S., J. Zhang, and C. Zhang (2022). Hyperspectral image denoising by low-rank models with hyper-laplacian total variation prior. Signal Processing 201, 108733.
  19. Yang, F. (2024). Linear spectral statistics of eigenvectors of anisotropic sample covariance matrices. Annales de l’Institut Henri Poincar´e, Probabilit´es et Statistiques 60(4), 2767– 2812.
  20. Yang, J., D. Zhang, A. F. Frangi, and J.-y. Yang (2004). Two-dimensional PCA: A new approach to appearance-based face representation and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(1), 131–137.
  21. Yuan, D. and N. Mancuso (2023). SuSiE PCA: A scalable Bayesian variable selection technique for principal component analysis. iScience 26(11), 108181.
  22. Zha, Z., X. Yuan, J. Zhou, C. Zhu, and B. Wen (2020). Image restoration via simultaneous nonlocal self-similarity priors. IEEE Transactions on Image Processing 29, 8561–8576.
  23. Zhang, H., H. Bi, X. Wang, and P. Zhang (2023). A joint-norm distance metric 2DPCA for robust dimensionality reduction. Information Sciences 640, 119036.
  24. Zhao, M., Z. Jia, Y. Cai, X. Chen, and D. Gong (2021). Advanced variations of two-dimensional principal component analysis for face recognition. Neurocomputing 452, 653–664.
  25. Zheng, S., Z. Bai, and J. Yao (2015). Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing. The Annals of Statistics 43(2), 546–591.
  26. Zhou, Y., J. Jiao, H. Huang, Y. Wang, J. Wang, H. Shi, and T. Huang (2020). When AWGNbased denoiser meets real noises. In Proceedings of the AAAI Conference on Artificial
  27. Intelligence, Volume 34, pp. 13074–13081.
  28. First author affiliation: School of Mathematics and Statistics, Xi’an Jiaotong University, No.28, Xianning West Road, Xi’an, Shaanxi, 710049, China.

Acknowledgments

The authors are grateful to the Editor, the Associate Editors, and the

referees for their review of the paper.

Dandan Jiang was supported by

NSFC under Grant No. 12571311.

Supplementary Materials

The online Supplementary Materials include: technical proofs for Lemmas 1–2 and Theorems 2–4; calculations for Example 1; preprocessing de-

tails; additional simulations; and facial database experiments, specifically

empirical support and analysis of the FERET dataset.

Supplementary materials are available for download.