﻿ 基于截断核范数张量鲁棒主成分分析

# 基于截断核范数张量鲁棒主成分分析Tensor Robust Principal Component Analysis Based on Truncated Nuclear Norm

Abstract: Low-tubal-rank tensor decomposition has been attracting attention of various fields due to the real application in image processing. However, conventional algorithms for tensor decomposition utilise the entire data to obtain the Low-tubal-rank and sparse components of a given tensor. Although many existing methods have fast convergence rates, these methods ignore the fact that small singular values contain little information. Based on this fact, we come up with a new decomposition method. Our method can simplify the tensor decomposition according to constrain the nuclear norm. Compared with the experimental results of many other tensor recovery methods, our proposed method can obtain a better effect.

1. 引言

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RPCA在多项式时间内仍然保持良好的恢复效果和稳定性，所以该方法被广泛地应用于数据分析工作中。遗憾的是，在大部分情况下RPCA只能处理矩阵数据，即二维数组。然而高维数据(即张量)在实际生活和科研工作中随处可见，比如彩色图片，高光谱图像大部分都被编译为三阶张量：行，列元素的分布以及每一个像素的颜色；彩色视频也可以视为四阶张量。张量在图像去噪 [6]，视频存储 [7]，数据挖掘 [8]，背景提取 [9] 中都有着非常广泛的应用。而对于张量形式的分解，具体来说，就是给定一个三阶张量，它可以被分解和被表示为，其中的空间内分别具有低秩和稀疏的结构。

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Figure 1. Illustration of matrix and tensor decomposition

2. 张量的相关定义与运算

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3. 主要模型及算法

3.1. 图像的低秩性

Figure 2. (a & A) Image “Lenna” with dimension 512 × 512 × 3 and its singular. (b & B) Image “Landscape” with dimension 3225×2491×3 and its singular. (c & C) Image “Sunflowers” with dimension 5272 × 3997 × 3 and its singular

3.2. 模型的建立与求解

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1) 固定变量，更新

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2) 固定变量，更新，类似的，我们有：

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3) 固定变量，更新

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4) 更新参数

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4. 实验结果与分析

4.1. 参数δ的取值

Figure 3. Recovery results of our method on image “Lenna” when corrupted rate, and singular value keeping rate varies from 0.2 to 0.95

4.2. 模拟实验结果及分析

Figure 4. Comparison of image recovery. (a) Original image; (b) corrupted image; (c)-(e) Recovered images by RPCA, SNN and Ours, respectively

Table 1. Recovering result by RPCA, SNN and our method (PSNR/SSIM/Time)

SNN对于张量的恢复效果也很不错，但是我们的方法得到的PSNR和SSIM还是比SNN来得高。并且从时间上来看，SNN恢复所花的时间会更加的漫长，几乎是我们方法的两倍。如果是更加高清的图片，张量规模会更大，我们方法在时间复杂度上的优势会更加明显。

5. 总结与展望

[1] Litjens, G., Kooi, T., Bejnordi, B.E., et al. (2017) A Survey on Deep Learning in Medical Image Analysis. Medical Image Analysis, 42, 60-88.
https://doi.org/10.1016/j.media.2017.07.005

[2] Brubaker, S.W., Bonham, K.S., Zanoni, I., et al. (2015) Innate Immune Pattern Recognition: A Cell Biological Perspective. Annual Review of Immunology, 33, 257-290.
https://doi.org/10.1146/annurev-immunol-032414-112240

[3] Hancock, P.J.B., Burton, A.M. and Bruce, V. (1996) Face Processing: Human Perception and Principal Components Analysis. Memory & Cognition, 24, 26-40.
https://doi.org/10.3758/BF03197270

[4] Misra, J., Schmitt, W., Hwang, D., et al. (2002) Interactive Exploration of Microarraygene Expression Patterns in a Reduced Dimensional Space. Genome Research, 12, 1112-1120.
https://doi.org/10.1101/gr.225302

[5] Candes, E., Li, X., Ma, Y. and Wright, J. (2011) Robust Principal Component Analysis? Journal of the ACM, 58, Article No. 11.
https://doi.org/10.1145/1970392.1970395

[6] Liu, J., Musialski, P., Wonka, P. and Ye, J. (2013) Tensor Completion for Estimating Missing Values in Visual Data. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35, 208-220.
https://doi.org/10.1109/TPAMI.2012.39

[7] Ji, H., Huang, S., Shen, Z. and Xu, Y. (2011) Robust Video Restoration by Joint Sparse and Low Rank Matrix Approximation. SIAM Journal on Imaging Sciences, 4, 1122-1142.
https://doi.org/10.1137/100817206

[8] Mørup, M. (2011) Applications of Tensor (Multiway Array) Factorizations and Decompositions in Data Mining. Data Mining and Knowledge Discovery, 1, 24-40.
https://doi.org/10.1002/widm.1

[9] Cao, W., Wang, Y., et al. (2016) Total Variation Regularized Tensor RPCA for Background Subtraction from Compressive Measurements. IEEE Transactions on Image Processing, 25, 4075-4090.
https://doi.org/10.1109/TIP.2016.2579262

[10] Kiers, H.A. (2000) Towards a Standardized Notation and Terminology in Multiway Analysis. Journal of Chemometrics: A Journal of the Chemometrics Society, 14, 105-122.
https://doi.org/10.1002/1099-128X(200005/06)14:3<105::AID-CEM582>3.0.CO;2-I

[11] Tucker, L.R. (1996) Some Mathematical Notes on Three-Mode Factor Analysis. Psychometrika, 31, 279-311.
https://doi.org/10.1007/BF02289464

[12] Hillar, C.J. and Lim, L.-H. (2013) Most Tensor Problems Are NP-Hard. Journal of the ACM, 60, 45.
https://doi.org/10.1145/2512329

[13] Lu, C., Feng, J., Chen, Y., Liu, W., Lin, Z. and Yan, S. (2016) Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization. 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, 27-30 June 2016, 5249-5257.
https://doi.org/10.1109/CVPR.2016.567

[14] 谢瑞, 王春祥, 马会阳, 张永显. 等式约束病态模型的截断奇异值解及其统计性质[J]. 测绘科学技术学报, 2019, 36(3): 227-232+237.

[15] 王艺卓, 曾海金, 赵佳佳, 谢晓振. 基于张量截断核范数的高光谱图像超分辨率重构[J]. 激光与光电子学进展, 2019, 56(21): 80-89.

[16] Kilmer, M.E. and Martin, C.D. (2011) Factorization Strategies for Third-Order Tensors. Linear Algebra and Its Applications, 435, 641-658.
https://doi.org/10.1016/j.laa.2010.09.020

[17] Zhang, Z., Ely, G., Aeron, S., Hao, N. and Kilmer, M. (2014) Novel Methods for Multilinear Data Completion and Denoising Based on Tensor-SVD. 2014 IEEE Conference on Computer Vision and Pattern Recognition, Columbus, OH, 23-28 June 2014, 3842-3849.
https://doi.org/10.1109/CVPR.2014.485

[18] Hu, Y., Zhang D.B., Ye, J.P., et al. (2013) 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] Huang, B., Mu, C., Goldfarb, D., et al. (2014) Provable Low-Rank Tensor Recovery. Optimization-Online, 4252, 455-500.

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