Principal Component Analysis (PCA) is a widely used technique for data analysis and dimensionality reduction, but its sensitivity to feature scale and outliers limits its applicability. Robust Principal Component Analysis (RPCA) addresses these limitations by decomposing data into a low-rank matrix capturing the underlying structure and a sparse matrix identifying outliers, enhancing robustness against noise and outliers. This paper introduces a novel RPCA variant, Robust PCA Integrating Sparse and Low-rank Priors (RPCA-SL). Each prior targets a specific aspect of the data’s underlying structure and their combination allows for a more nuanced and accurate separation of the main data components from outliers and noise. Then RPCA-SL is solved by employing a proximal gradient algorithm for improved anomaly detection and data decomposition. Experimental results on simulation and real data demonstrate significant advancements.
References
[1]
Jolliffe, I.T. (2002) Principal Component Analysis. 2nd Edition, Springer-Verlag, Berlin.
[2]
Abdi, H. and Williams, L.J. (2010) Principal Component Analysis. Wiley Interdisciplinary Reviews: Computational Statistics, 2, 433-459. https://doi.org/10.1002/wics.101
[3]
Ringnér, M. (2008) What Is Principal Component Analysis? Nature Biotechnology, 26, 303-304. https://doi.org/10.1038/nbt0308-303
[4]
Jafarzadegan, M., Safi-Esfahani, F. and Beheshti, Z. (2019) Combining Hierarchical Clustering Approaches Using the PCA Method. Expert Systems with Applications, 137, 1-10. https://doi.org/10.1016/j.eswa.2019.06.064
[5]
Wang, Q., Gao, Q.X., Sun, G., et al. (2020) Double Robust Principal Component Analysis. Neurocomputing, 391, 119-128. https://doi.org/10.1016/j.neucom.2020.01.097
[6]
Candes, E.J., Li, X., Ma, Y. and Wright, J. (2011) Robust Principal Component Analysis? Journal of the ACM, 58, 1-37. https://doi.org/10.1145/1970392.1970395
[7]
Vaswani, N., Bouwmans, T., Javed, S. and Narayanamurthy, P. (2018) Robust Subspace Learning: Robust PCA Robust Subspace Tracking and Robust Subspace Recovery. IEEE Signal Processing Magazine, 35, 32-55. https://doi.org/10.1109/MSP.2018.2826566
[8]
Lyu, Q., Guo, M., Ma, M., et al. (2019) External Prior Learning and Internal Mean Sparse Coding for Image Denoising. Journal of Electronic Imaging, 28, 1-10. https://doi.org/10.1117/1.JEI.28.3.033014
[9]
Wang, Z., Li, X., So, H.C. and Liu, Z. (2022) Robust PCA via Non-Convex Half-Quadratic Regularization. Signal Processing, 204, Article 108816. https://doi.org/10.1016/j.sigpro.2022.108816
[10]
Bouwmans, T. and Zahzah, E.H. (2014) Robust PCA via Principal Component Pursuit: A Review for a Comparative Evaluation in Video Surveillance. Computer Vision and Image Understanding, 122, 22-34. https://doi.org/10.1016/j.cviu.2013.11.009
[11]
Hu, Z., Wang, Y., Su, R., Bian, X., Wei, H. and He, G. (2020) Moving Object Detection Based on Non-Convex RPCA with Segmentation Constraint. IEEE Access, 8, 41026-41036. https://doi.org/10.1109/ACCESS.2020.2977273
[12]
Shijila, B., Tom, A.J. and George, S.N. (2018) Moving Object Detection by Low Rank Approximation and l1-TV Regularization on RPCA Framework. Journal of Visual Communication and Image Representation, 56, 188-200. https://doi.org/10.1016/j.jvcir.2018.09.009
[13]
Liu, J. and Osher, S. (2019) Block Matching Local SVD Operator Based Sparsity and TV Regularization for Image Denoising. Journal of Scientific Computing, 78, 607-624. https://doi.org/10.1007/s10915-018-0785-8
[14]
Chen, Y., Xiao, X. and Zhou, Y. (2020) Low-Rank Quaternion Approximation for Color Image Processing. IEEE Transactions on Image Processing, 29, 1426-1439. https://doi.org/10.1109/TIP.2019.2941319
[15]
Chen, Y., Wang, S. and Zhou, Y. (2018) Tensor Nuclear Norm-Based Low-Rank Approximation with Total Variation Regularization. IEEE Journal of Selected Topics in Signal Processing, 12, 1364-1377. https://doi.org/10.1109/JSTSP.2018.2873148
[16]
Mousavi, A., Rezaee, M., Ayanzadeh, R. (2020) A Survey on Compressive Sensing: Classical Results and Recent Advancements. Journal of Mathematical Modeling, 8, 309-344.
[17]
Feng, L. and Wang, J. (2022) Projected Robust PCA with Application to Smooth Image Recovery. Journal of Machine Learning Research, 23, 1-41. https://dl.acm.org/doi/pdf/10.5555/3586589.3586838
[18]
Hu, Y., Zhang, D., Ye, J., Li, X. and He, X. (2012) Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35, 2117-2130. https://doi.org/10.1109/TPAMI.2012.271
[19]
Zhao, Q., Lin, Y., Wang, F. and Meng, D. (2024) Adaptive Weighting Function for Weighted Nuclear Norm Based Matrix/Tensor Completion. International Journal of Machine Learning and Cybernetics, 15, 697-718. https://doi.org/10.1007/s13042-023-01935-1
[20]
Cai, H.Q., Hamm, K., Huang, L., Li, J. and Wang, T. (2020) Rapid Robust Principal Component Analysis: CUR Accelerated Inexact Low Rank Estimation. IEEE Signal Processing Letters, 28, 116-120. https://doi.org/10.1109/LSP.2020.3044130
[21]
Antonello, N., Stella, L., Patrinos, P. and Van Waterschoot, T. (2018) Proximal Gradient Algorithms: Applications in Signal Processing. ar-Xiv:1803.01621
[22]
Parikh, N. and Boyd, S. (2014) Proximal Algorithms. Foundations and Trends® in Optimization, 1, 127-239. https://doi.org/10.1561/2400000003
[23]
Cai, H., Cai, J.-F. and Wei, K. (2019) Accelerated Alternating Projections for Robust Principal Component Analysis. Journal of Machine Learning Research, 20, 685-717.
[24]
Li, L., Huang, W., Gu, I.Y.H., et al. (2004) Statistical Modeling of Complex Backgrounds for Foreground Object Detection. IEEE Transactions on Image Processing, 13, 1459-1472. https://doi.org/10.1109/TIP.2004.836169
[25]
Yasuma, F., Mitsunaga, T., Iso, D. and Nayar. S.K. (2008) Generalized Assorted Pixel Camera: Post-Capture Control of Resolution, Dynamic Range and Spectrum. Technical Report, Department of Computer Science, Columbia University, New York.
