# 基于秩的逼近的张量鲁棒主成分分析Tensor Robust Principal Component Analysis via Non-Convex Rank Approximation

Abstract: In this paper, a new tensor rank approximation is proposed and a non-convex TRPCA model is established. An effective augmented Lagrangian multiplier optimization algorithm is used to solve this non-convex minimization problem. The experimental results show that compared with the tensor robust principal component analysis algorithm based on kernel norm, the estimated tensor obtained by this algorithm has smaller deviation and is effective in terms of accuracy and efficiency.

1. 引言

2. 预备知识

$bdiag\left(\stackrel{¯}{A}\right)=\left[\begin{array}{cccc}{\stackrel{¯}{A}}^{\left(1\right)}& & & \\ & {\stackrel{¯}{A}}^{\left(2\right)}& & \\ & & \ddots & \\ & & & {\stackrel{¯}{A}}^{\left({n}_{3}\right)}\end{array}\right]\in {C}^{{n}_{1}{n}_{3}×{n}_{2}{n}_{3}}$

$bcirc\left(\mathcal{A}\right)=\left[\begin{array}{cccc}{A}^{\left(1\right)}& {A}^{\left({n}_{3}\right)}& \cdots & {A}^{\left(2\right)}\\ {A}^{\left(2\right)}& {A}^{\left(1\right)}& \cdots & {A}^{\left(3\right)}\\ ⋮& ⋮& \ddots & ⋮\\ {A}^{\left({n}_{3}\right)}& {A}^{\left({n}_{3}-1\right)}& \cdots & {A}^{\left(1\right)}\end{array}\right]\in {C}^{{n}_{1}{n}_{3}×{n}_{2}{n}_{3}}$

$\stackrel{¯}{A}=fft\left(A,\left[\right],3\right)$

$A=U\ast S\ast {V}^{\ast }$

3. 通过秩的非凸逼近的低秩张量RPCA算法

${‖L‖}_{\gamma }=\frac{1}{{n}_{3}}\underset{i}{\sum }\underset{j}{\sum }\frac{\left(1+\gamma \right){\sigma }_{i}\left({\stackrel{¯}{L}}^{\left(j\right)}\right)}{\gamma +{\sigma }_{i}\left({\stackrel{¯}{L}}^{\left(j\right)}\right)}$ (3.1)

3.2张量RPCA的非凸优化模型

$\underset{L,S}{\mathrm{min}}{‖L‖}_{\gamma }+{‖S‖}_{{l}_{1}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{s}\text{.t}\text{.}\text{\hspace{0.17em}}X=L+S$ (3.2)

$L\left(L,S,Y,\mu \right)={‖L‖}_{\gamma }+\lambda {‖S‖}_{{l}_{1}}+〈Y,L+S-X〉+\frac{\mu }{2}{‖L+S-X‖}_{F}^{2}$ (3.3)

${L}_{k+1}=\underset{L}{\mathrm{arg}\mathrm{min}}{‖L‖}_{\gamma }+\frac{{\mu }^{k}}{2}{‖L-\left(X-{S}_{k}-\frac{{Y}_{k}}{{\mu }_{k}}\right)‖}_{F}^{2}$ (3.4)

${S}_{k+1}=\underset{S}{\mathrm{arg}\mathrm{min}}\lambda {‖S‖}_{{l}_{1}}+\frac{{\mu }_{k}}{2}{‖S-\left(X-{L}_{k+1}-\frac{{Y}_{k}}{{\mu }_{k}}\right)‖}_{F}^{2}$ (3.5)

${Y}_{k+1}={Y}_{k}+{\mu }_{k}\left({L}^{k+1}-X+{S}^{k+1}\right)$ (3.6)

${\mu }_{k+1}=\rho {\mu }_{k}$ (3.7)

$\underset{X}{\mathrm{min}}F\left(X\right)+\frac{\mu }{2}{‖X-A‖}_{F}^{2}$

$pro{x}_{f,\mu }\left({\sigma }_{A}\right)\underset{\sigma }{=\mathrm{arg}\mathrm{min}}f\left(\sigma \right)+\frac{\mu }{2}{‖\sigma -{\sigma }_{A}‖}_{2}^{2}$

${\sigma }_{k+1}=\mathrm{arg}\mathrm{min}〈{w}_{k},\sigma 〉+\frac{{\mu }^{t}}{2}{‖\sigma -{\sigma }_{A}‖}_{2}^{2}$

${‖\stackrel{^}{L}‖}_{\gamma }=\underset{i}{\sum }\frac{\left(1+\gamma \right){\sigma }_{i}\left(\stackrel{^}{L}\right)}{\gamma +{\sigma }_{i}\left(\stackrel{^}{L}\right)}=\underset{j}{\sum }\underset{i}{\sum }\frac{\left(1+\gamma \right){\sigma }_{i}\left({\stackrel{¯}{L}}^{\left(j\right)}\right)}{\gamma +{\sigma }_{i}\left({\stackrel{¯}{L}}^{\left( j \right)}\right)}$

$\mathrm{min}\frac{{\mu }^{k}}{2}\cdot \frac{1}{{n}_{3}}{‖\stackrel{^}{L}-\stackrel{^}{Q}‖}_{F}^{2}+\frac{1}{{n}_{3}}\cdot {‖\stackrel{^}{L}‖}_{\gamma }$ (3.8)

$\begin{array}{c}{\left(\stackrel{^}{L}\right)}^{k+1}=\underset{\stackrel{^}{L}}{\mathrm{arg}\mathrm{min}}\frac{{\mu }^{k}}{2}\cdot \frac{1}{{n}_{3}}{‖\stackrel{^}{L}-\stackrel{^}{Q}‖}_{F}^{2}+\frac{1}{{n}_{3}}\cdot {‖\stackrel{^}{L}‖}_{\gamma }\\ =\frac{1}{{n}_{3}}\underset{\stackrel{^}{L}}{\mathrm{arg}\mathrm{min}}\left\{\frac{{\mu }_{k}}{2}{‖\stackrel{^}{L}-\stackrel{^}{Q}‖}_{F}^{2}+{‖\stackrel{^}{L}‖}_{\gamma }\right\}\end{array}$ (3.9)

$pro{x}_{f,\mu }\left({\sigma }_{\stackrel{^}{Q}}\right)\underset{\sigma }{=\mathrm{arg}\mathrm{min}}f\left(\sigma \right)+\frac{\mu }{2}{‖\sigma -{\sigma }_{\stackrel{^}{Q}}‖}_{2}^{2}$ (3.10)

${\sigma }_{k+1}=\mathrm{arg}\mathrm{min}〈{w}_{k},\sigma 〉+\frac{{\mu }^{t}}{2}{‖\sigma -{\sigma }_{\stackrel{^}{Q}}‖}_{2}^{2}$ (3.11)

${\sigma }_{k+1}={\left({\sigma }_{\stackrel{^}{Q}}-\frac{{w}_{k}}{{\mu }^{t}}\right)}_{+}=\mathrm{max}\left({\sigma }_{\stackrel{^}{Q}}-\frac{{w}_{k}}{{\mu }^{t}},0\right)$ (3.12)

4. 图像数据实验

Figure 1. Image recovery comparison by using different methods (a) Original image (b) Noisy images. Recovered image by (c) TRPCA (d) SNN (e) NCTRPCA, respectively

5. 结论

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https://doi.org/10.1109/ICDM.2015.15

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https://doi.org/10.1561/2200000016

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

[4] Tao, P.D. and An, L.T.H. (1997) Convex Analysis Approach to DC Programming: Theory Algorithms and Applications. Acta Mathematica Vietnamica, 22, 289-355.

[5] 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 Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Las Vegas, 27-30 June 2016, 5249-5257.
https://doi.org/10.1109/CVPR.2016.567

[6] Huang, B., Mu, C., Goldfarb, D. and Wright, J. (2014) Provable Low-Rank Tensor Recovery. Optimization-Online, 4252.

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