# 基于截断核范数的张量去噪Tensor Denoising Based on Truncated Nuclear Norm

Abstract: Based on the problem of tensor nuclear norm, a new robust principal component analysis model is established by substituting truncated nuclear norm for nuclear norm, and the convex optimization problem is solved by augmented Lagrange multiplier method. In the experiment of image denoising, the robust principal component analysis model with truncated unclear norm has good denosing effect.

1. 引言

2. 预备知识

${‖\mathcal{A}‖}_{\ast }:=\frac{1}{{n}_{3}}{\sum }_{i=1}^{{n}_{3}}{‖{\stackrel{¯}{A}}^{\left(i\right)}‖}_{\ast }$ (1)

${‖\mathcal{X}‖}_{r}={‖\mathcal{X}‖}_{\ast }-\underset{\begin{array}{l}\mathcal{A}\ast {\mathcal{A}}^{\text{T}}=\mathcal{I}\\ \mathcal{B}\ast {\mathcal{B}}^{\text{T}}=\mathcal{I}\end{array}}{\mathrm{max}}tr\left(\mathcal{A}\ast \mathcal{X}\ast \mathcal{B}\right)$ (2)

$\underset{i=1,\cdots ,{n}_{1}}{\mathrm{max}}{‖{\mathcal{U}}^{\ast }\ast \stackrel{\circ }{{\mathfrak{e}}_{i}}‖}_{F}\le \sqrt{\frac{\mu r}{{n}_{1}{n}_{3}}}$ (3)

$\underset{j=1,\cdots ,{n}_{2}}{\mathrm{max}}{‖{\mathcal{V}}^{\ast }\ast \stackrel{\circ }{{\mathfrak{e}}_{j}}‖}_{F}\le \sqrt{\frac{\mu r}{{n}_{2}{n}_{3}}}$ (4)

${‖\mathcal{U}\ast {\mathcal{V}}^{\ast }‖}_{\infty }\le \sqrt{\frac{\mu r}{{n}_{1}{n}_{2}{n}_{2}^{3}}}$ (5)

$\mathcal{A}=\mathcal{U}\ast \mathcal{S}\ast {\mathcal{V}}^{\ast }$ (6)

$\mathcal{U}\in {ℝ}^{{n}_{1}×{n}_{1}×{n}_{3}},\mathcal{V}\in {ℝ}^{{n}_{2}×{n}_{2}×{n}_{3}}$ 是正交的， $\mathcal{S}\in {ℝ}^{{n}_{1}×{n}_{2}×{n}_{3}}$ 是f-对角张量。

(7)

3. 截断核范数的鲁棒主成分分析

3.1. 模型的建立

(8)

(9)

3.2. 模型的求解

$\underset{X}{\mathrm{min}}\text{\hspace{0.17em}}\text{ }{‖X‖}_{\ast }-Tr\left({\mathcal{A}}_{l}\mathcal{W}{\mathcal{B}}_{l}^{\text{T}}\right)+\lambda {‖\mathcal{E}‖}_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{ }\mathcal{L}=\mathcal{W},\text{\hspace{0.17em}}\mathcal{X}=\mathcal{L}+\mathcal{E}$ (10)

$\begin{array}{c}L\left(\mathcal{L},\mathcal{E},\mathcal{W},{\mathcal{Y}}_{1},{\mathcal{Y}}_{2}\right)={‖\mathcal{L}‖}_{\ast }-Tr\left({\mathcal{A}}_{l}\mathcal{W}{\mathcal{B}}_{l}^{\text{T}}\right)+\lambda {‖\mathcal{E}‖}_{1}+〈{\mathcal{Y}}_{1},\mathcal{L}+\mathcal{E}-\mathcal{X}〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+〈{\mathcal{Y}}_{2},\mathcal{L}-\mathcal{W}〉+\frac{\mu }{2}\left({‖\mathcal{L}+\mathcal{E}-\mathcal{X}‖}_{F}^{2}+{‖\mathcal{L}-\mathcal{W}‖}_{F}^{2}\right)\end{array}$ (11)

$\begin{array}{c}{\mathcal{L}}_{k+1}=\mathrm{arg}\underset{\mathcal{L}}{\mathrm{min}}\text{\hspace{0.17em}}{‖\mathcal{L}‖}_{\ast }+〈{\mathcal{Y}}_{1,k},\mathcal{L}+{\mathcal{E}}_{k}-\mathcal{X}〉+〈{\mathcal{Y}}_{2,k},\mathcal{L}-{\mathcal{W}}_{k}〉+\frac{\mu }{2}\left({‖\mathcal{L}+{\mathcal{E}}_{k}-\mathcal{X}‖}_{F}^{2}+{‖\mathcal{L}-{\mathcal{W}}_{k}‖}_{F}^{2}\right)\\ =\mathrm{arg}\underset{\mathcal{L}}{\mathrm{min}}\text{\hspace{0.17em}}{‖\mathcal{L}‖}_{\ast }+\mu {‖\mathcal{L}+\frac{1}{2}\left({\mathcal{E}}_{k}-\mathcal{X}-{\mathcal{W}}_{k}+\frac{1}{\mu }\left({\mathcal{Y}}_{1k}+{\mathcal{Y}}_{2k}\right)\right)‖}_{F}^{2}\end{array}$ (12)

(13)

$\begin{array}{c}{\mathcal{E}}_{k+1}=\mathrm{arg}\underset{\mathcal{E}}{\mathrm{min}}\lambda {‖\mathcal{E}‖}_{1}+〈{\mathcal{Y}}_{1k},{\mathcal{L}}_{k+1}+\mathcal{E}-\mathcal{X}〉+\frac{\mu }{2}{‖{\mathcal{L}}_{k+1}+\mathcal{E}-\mathcal{X}‖}_{F}^{2}\\ =\mathrm{arg}\underset{\mathcal{E}}{\mathrm{min}}\lambda {‖\mathcal{E}‖}_{1}+\frac{\mu }{2}{‖\mathcal{E}+\left({\mathcal{L}}_{k+1}-\mathcal{X}+\frac{1}{\mu }{\mathcal{Y}}_{1,k}\right)‖}_{F}^{2}\end{array}$ (14)

${\mathcal{E}}_{k+1}={\mathcal{S}}_{\frac{\lambda }{{\mu }_{k}}}\left(\mathcal{X}-{\mathcal{L}}_{k+1}-\frac{1}{\mu }{\mathcal{Y}}_{1,k}\right)$ (15)

${\mathcal{W}}_{k+1}=\mathrm{arg}\underset{\mathcal{W}}{\mathrm{min}}\text{\hspace{0.17em}}\text{ }-Tr\left({\mathcal{A}}_{l}\mathcal{W}{\mathcal{B}}_{l}^{\text{T}}\right)+〈{\mathcal{Y}}_{2k},{\mathcal{L}}_{k+1}-\mathcal{W}〉+\frac{\mu }{2}{‖{\mathcal{L}}_{k+1}-\mathcal{W}‖}_{F}^{2}$ (16)

(16)式为关于 $\mathcal{W}$ 的二次项，令其导数为0，可得到闭式解。去掉无关项得：

${\mathcal{W}}_{k+1}=\mathrm{arg}\underset{\mathcal{W}}{\mathrm{min}}\text{\hspace{0.17em}}\text{ }-Tr\left({\mathcal{A}}_{l}\mathcal{W}{\mathcal{B}}_{l}^{\text{T}}\right)+\frac{\mu }{2}{‖\mathcal{W}‖}_{F}^{2}+〈-{\mathcal{Y}}_{2k},\mathcal{W}〉+\mu 〈-{\mathcal{L}}_{k+1},\mathcal{W}〉$ (17)

$-{\mathcal{A}}_{l}{\mathcal{B}}_{l}^{\text{T}}+\mu \mathcal{W}-{\mathcal{Y}}_{2k}-\mu {\mathcal{L}}_{k+1}=0$ (18)

$\mathcal{W}={\mathcal{L}}_{k+1}+\frac{1}{\mu }\left({\mathcal{A}}_{l}{\mathcal{B}}_{l}^{\text{T}}+{\mathcal{Y}}_{2k}\right)$ (19)

$\begin{array}{l}{\mathcal{Y}}_{1k+1}={\mathcal{Y}}_{1k}+\mu \left({\mathcal{L}}_{k+1}+{\mathcal{E}}_{k+1}-\mathcal{X}\right)\\ {\mathcal{Y}}_{2k+1}={\mathcal{Y}}_{2k}+\mu \left({\mathcal{L}}_{k+1}-{\mathcal{W}}_{k+1}\right)\end{array}$ (20)

4. 实验

Figure 1. Comparison of denoising effects of different methods

Table 1. Data denoised by each algorithm

5. 结论

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[7] 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, NV, 27-30 June 2016, 5249-5257.
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[8] Xue, S., Qiu, W., Liu, F. and Jin, X. (2017) Low-Rank Tensor Completion by Truncated Nuclear Norm Regularization. arXiv preprint arXiv:1712.00704
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