CUR 分解是一種具高度可解釋性的資料分析工具，其目標為利用原始資料矩陣中少數的行與列來進行近似，從而在保留原始資料結構的同時，加速大規模矩陣的運算效率。本文旨在改進 Xu 等人所提出之基於能量的自適應CUR 分解方法，提出一種基於快速傅立葉轉換的自適應 CUR 分解演算法。首先，我們回顧 CUR 分解的基本架構及其所涉及的行選取問題。接著，我們運用快速傅立葉轉換的頻域特性，結合 Wang 與 Zhang 所提出的自適應取樣方法，提出一種高效的行選取流程，進而提升 CUR 分解在取樣品質與計算效率上的表現，這也是本文最主要的貢獻。最後，我們透過數值模擬實驗驗證本方法於滿秩與非滿秩影像矩陣上的適用性，結果顯示其在精確度與效能方面皆具顯著優勢。

CUR decomposition approximates a data matrix using a small number of its actual columns and rows, making it a highly interpretable data analysis tool. Its primary objective is to accelerate large-scale matrix computations while preserving the structural information of the original data. This thesis aims to improve the energy-based adaptive CUR decomposition method proposed by Xu et al. by introducing an adaptive CUR decomposition algorithm based on the Fast Fourier Transform (FFT). We first review the fundamental framework of CUR decomposition and the associated column selection problem. Then, leveraging the frequency-domain characteristics of FFT and integrating the adaptive sampling strategy proposed by Wang and Zhang, we design an efficient column selection procedure to enhance both the sampling quality and computational efficiency of CUR decomposition. This contribution constitutes the core innovation of this work. Finally, we validate the applicability of the proposed method through numerical experiments on both full-rank and non-full-rank image matrices, with results demonstrating significant improvements in both accuracy and performance.

[1] M. W. Mahoney and P. Drineas, CUR matrix decompositions for improved data analysis, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), pp. 697–702.
[2] A. Aldroubi, K. Hamm, A. B. Koku, and A. Sekmen, CUR decompositions, similarity matrices, and subspace clustering, Frontiers in Applied Mathematics and Statistics, 4 (2019), pp. 1-17.
[3] S. Khan, M. Nawaz, X. Guoxia and H. Yan, Image correspondence with CUR decomposition based graph completion and matching, IEEE Transactions on Circuits and Systems for Video Technology, 30 (2020), pp. 3054-3067.
[4] H. Q. Cai, K. Hamm, L. Huang, J. Li and T. Wang, Rapid robust principal component analysis: CUR accelerated inexact low-rank estimation, IEEE Signal Processing Letters, 28 (2021), pp. 116-120.
[5] N. Yadav, N. Monath, R. Angell, M. Zaheer and A. McCallum, Efficient nearest neighbor search for Cross-Encoder models using matrix factorization, Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing (EMNLP), 2022, pp. 2171-2194.
[6] A. Deshpande and S. Vempala, Adaptive sampling and fast low-rank matrix approximation, Proceedings of the 10th International Workshop on Randomization and Approximation Techniques in Computer Science (APPROX), 2006, pp. 292-303.
[7] P. Drineas, M. W. Mahoney, and S. Muthukrishnan, Relative-error CUR matrix decompositions, SIAM Journal on Matrix Analysis and Applications, 30 (2008), pp. 844-881.
[8] C. Boutsidis, P. Drineas, and M. Magdon-Ismail, Near-optimal column-based matrix reconstruction, SIAM Journal on Computing, 43 (2014), pp. 687-717.
[9] S. Wang and Z. Zhang, Improving CUR matrix decomposition and the Nyström approximation via adaptive sampling, Journal of Machine Learning Research, 14 (2013), pp. 2549-2589.
[10] S. Wang, Z. Zhang, and T. Zhang, Towards more efficient SPSD matrix approximation and CUR matrix decomposition, Journal of Machine Learning Research, 17 (2016), pp. 1-49.
[11] L.-W. Xu, X.-J. Zhao, and Y.-X. Zhang, Energy-based adaptive CUR matrix decomposition, IEEE Access, 11 (2023), pp. 21934-21944.
[12] K. Hamm, L.-X. Huang, Perspectives on CUR decompositions, Applied and Computational Harmonic Analysis, 48 (2020), pp. 1088-1099.
[13] J. W. Cooley and J. W. Tukey, An algorithm for the machine calculation of complex Fourier series, Mathematics of Computation, 19 (1965), pp. 297-301.
[14] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 2nd ed., Prentice Hall, (1999).
[15] D. Crescini, A. Flammini, D. Marioli and A. Taroni, Application of an FFT based algorithm to signal processing of LVDT position sensors, IEEE Transactions on Instrumentation and Measurement, 47 (1998), pp. 1119-1124.
[16] L. Marple, Computing the discrete-time “analytic” signal via FFT, IEEE Transactions on Signal Processing, 47 (1999), pp. 2600-2603.
[17] A. S. Samra, S. E. GadAllah, and R. M. Ibrahim, Face recognition using wavelet transform, Fast Fourier Transform and discrete cosine transform, Proceedings of the 46th IEEE Midwest Symposium on Circuits and Systems(MWSCAS), 1 (2003), pp. 272-275.
[18] I. A. Ismaili, S. A. Khowaja and W. J. Soomro, Image compression, comparison between discrete cosine transform and Fast Fourier Transform and the problems associated with DCT, Proceedings of the International Conference on Image Processing and Computer Vision (IPC), 1 (2013).
[19] B. S. Reddy and B. N. Chatterji, An FFT-based technique for translation, rotation, and scale-invariant image registration, IEEE Transactions on Image Processing, 5 (1996), pp. 1266-1271.
[20] M. Frigo and S. G. Johnson, FFTW: an adaptive software architecture for the FFT, Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 3 (1998), pp. 1381-1384.