﻿ 基于加权Schatten-1/2范数的低秩矩阵近似算法

# 基于加权Schatten-1/2范数的低秩矩阵近似算法Weighted Schatten-1/2 Norm Minimization for Low-Rank Matrix Approximation

Abstract: In this paper, the low-rank matrix approximation problem is discussed with a weighted Schatten quasi-norm as the objective function, constrained by partial obtained data. The weights are in-troduced to measure the importance of different rank components. A weighted fixed point iterative thresholding algorithm is proposed based on the fixed point representation theory. The con-vergence analysis of the algorithm is provided. Numerical examples illustrate the effciency of our method.

1. 引言

$\begin{array}{l}\mathrm{min}\text{\hspace{0.17em}}\text{\hspace{0.17em}}rank\left(X\right)\\ s.t.\text{ }{X}_{i,j}={M}_{i,j},\text{}\forall \left(i,j\right)\in \Omega \end{array}$ (1.1)

$X\in {R}^{m×n}$$\Omega$ 表示待恢复矩阵M中可观测元素下标的集合。Candės分析了问题(1.1)的计算复杂度并证明其为NP-hard问题。随后，作为rank函数的凸包络，核范数被提出用来逼近问题(1.1)的目标函数

$\begin{array}{l}\underset{X}{\mathrm{min}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{‖X‖}_{\text{*}}\\ s.t.\text{ }{X}_{i,j}={M}_{i,j},\text{}\forall \left(i,j\right)\in \Omega \end{array}$ (1.2)

$\begin{array}{l}\underset{X}{\mathrm{min}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{‖X‖}_{p}^{p}\\ s.t.\text{ }{X}_{i,j}={M}_{i,j},\text{}\forall \left(i,j\right)\in \Omega \end{array}$ (1.3)

${‖X‖}_{p}={\left({\sum }_{k}{\sigma }_{k}^{p}\right)}^{\frac{1}{p}}$。问题(1.3)可由MM (Majorization-Minimization)迭代算法 [14] 计算数值解。然而，由于目标函数非凸、非光滑、非Lipschitz连续的性质，使得解析解难以求解 [13] [15] [16] [17] [18]。另一方面，参数p的变化也会对问题(1.3)产生很大影响。文献 [19] 指出当 $p\in \left[\frac{1}{2},1\right)$，近似结果相对理想，当 $p\in \left[0,\frac{1}{2}\right)$

$\begin{array}{l}\mathrm{min}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{‖X‖}_{W,{S}_{p}}^{p}\\ s.t.\text{ }{X}_{i,j}={M}_{i,j},\text{}\forall \left(i,j\right)\in \Omega \end{array}$ (1.4)

$〈X,Y〉=trace\left({Y}^{\text{T}}X\right)$$trace\left(X\right)={\sum }_{i}{X}_{i,i}$${‖X‖}_{F}={\left({\sum }_{i,j}{X}_{i,j}^{2}\right)}^{\frac{1}{2}}={\left({\sum }_{i=1}^{n}{\sigma }_{i}^{2}\right)}^{\frac{1}{2}}$。向量 $x,y\in {R}^{n}$${‖x‖}_{2}={\left({\sum }_{i=1}^{n}{x}_{i}^{2}\right)}^{\frac{1}{2}}$

$〈x,y〉={x}^{\text{T}}y$

2. 全局必要最优条件

$p=\frac{1}{2}$，给出加权 ${S}_{\frac{1}{2}}$ 拟范数正则化模型和相应的不动点迭代算法。

$\underset{X}{\mathrm{min}}\frac{1}{2}{‖{P}_{\Omega }\left(X\right)-{P}_{\Omega }\left(Y\right)‖}_{F}^{2}+{‖X‖}_{W,{S}_{\frac{1}{2}}}^{\frac{1}{2}}$ (2.1)

$A\text{}=\text{}U{\sum }_{A}V\text{},\text{}B\text{}=\text{}U{\sum }_{B}V$ (2.2)

${\Sigma }_{A},\text{}{\Sigma }_{B}$ 分别为矩阵 $A,B$ 的奇异值矩阵。

$\begin{array}{l}\underset{{\delta }_{1},\cdots ,{\delta }_{r}}{\mathrm{min}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sum }_{i}^{r}\left[{\left({\delta }_{i}-{\sigma }_{i}\right)}^{2}+{\omega }_{i}{\delta }_{i}^{\frac{1}{2}}\right]\\ s.t.\text{ }{\delta }_{i}\ge 0,\text{}{\delta }_{i}\ge {\delta }_{j},\text{}i\le j,\text{}i=1,\cdots ,r\text{}\end{array}$ (2.3)

${h}_{\lambda }\left(y\right)=\underset{x\ge 0}{\mathrm{arg}\mathrm{min}}\left\{{f}_{\lambda }\left(x\right)\right\}$ (2.4)

${h}_{\lambda }\left(y\right)=\left\{\begin{array}{c}{h}_{\lambda }\left(y\right),\text{}y>\frac{3}{4}{\lambda }^{\frac{2}{3}}\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0,\text{}y>\frac{3}{4}{\lambda }^{\frac{2}{3}}\end{array}$ (2.5)

${h}_{\lambda }\left(y\right)=\frac{2}{3}y\left(1+\mathrm{cos}\left(\frac{2\pi }{3}-\frac{2}{3}{\Phi }_{\lambda }\left(y\right)\right)\right)$ (2.6)

${\Phi }_{\lambda }\left(y\right)=\mathrm{arccos}\left(\frac{\lambda }{8}{\left(\frac{y}{3}\right)}^{-\frac{3}{2}}\right)$ (2.7)

${y}_{i}-{x}_{i}+\lambda \frac{sign\left({y}_{i}\right)}{4\sqrt{|{y}_{i}|}}=0$ (2.8)

$\sqrt{|{y}_{i}|}=\eta ,\text{}{y}_{i}={\eta }^{2}$。当 ${x}_{i}>\frac{3}{4}{\lambda }^{\frac{2}{3}}$ 时，有

${\eta }^{3}-{x}_{i}\eta +\frac{\lambda }{4}=0$ (2.9)

${\eta }_{2}=2r\mathrm{cos}\left(\frac{\pi }{3}+\frac{\Phi }{3}\right),$ ${\eta }_{3}=2r\mathrm{cos}\left(\frac{\pi }{3}-\frac{\Phi }{3}\right)$ [11]。经验证， ${\eta }_{3}=2r\mathrm{cos}\left(\frac{\pi }{3}-\frac{\Phi }{3}\right)$ 是问题(2.4)的最优解，且解

${y}_{i}=\frac{2}{3}{x}_{i}\left(1+\mathrm{cos}\left(\frac{2\pi }{3}-\frac{2}{3}\lambda {\Phi }_{\lambda }\left({x}_{i}\right)\right)\right)$ (2.10)

${x}_{i}>\frac{3}{4}{\lambda }^{\frac{2}{3}}$ 时，证明与上述类似。

${H}_{\lambda }\left(x\right):=\left({h}_{\lambda }\left({x}_{1}\right),{h}_{\lambda }\left({x}_{2}\right),\cdots ,{h}_{\lambda }\left({x}_{n}\right)\right),\text{}\forall x=\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)\in {R}^{n}$ (2.11)

${h}_{\lambda }\left(x\right)$ 的形式如(2.5)所示。

$\forall \lambda >0$，由向量阈值函数可以定义矩阵阈值函数

${Η}_{\lambda }\left(Y\right):=U{\sum }_{{H}_{\lambda }}{V}^{\text{T}}$ (2.12)

${\Sigma }_{{H}_{\lambda }}=diag\left({H}_{\lambda }\left({\sigma }_{1}\right),\cdots ,{H}_{\lambda }\left({\sigma }_{r}\right)\right)$。因此，问题(2.1)可以由矩阵阈值函数求解。本节的后续内容将会介绍由(2.12)表示的问题(2.1)解的形式以及权重对奇异值的影响。

${f}_{\delta }\left(x\right):={\left(\sigma -\delta \right)}^{2}+\omega {\delta }^{\frac{1}{2}}$ (2.13)

$y<\frac{3}{4}{\omega }_{i}^{\frac{2}{3}}$$y<\frac{3}{4}{\omega }_{j}^{\frac{2}{3}}$ 时，由(2.5)，有 ${h}_{{\omega }_{i}}={h}_{{\omega }_{j}}=0$。结论 ${h}_{{\omega }_{i}}\ge {h}_{{\omega }_{j}}$ 成立。

$y>\frac{3}{4}{\omega }_{i}^{\frac{2}{3}}$$y<\frac{3}{4}{\omega }_{j}^{\frac{2}{3}}$ 时，显然有 ${h}_{{\omega }_{j}}=0$。另一方面，由引理3的证明，当 $y>0$ 时有 ${h}_{{\omega }_{i}}>0$。因此，结论 ${h}_{{\omega }_{i}}\ge {h}_{{\omega }_{j}}$ 成立。

$y>\frac{3}{4}{\omega }_{i}^{\frac{2}{3}}$$y>\frac{3}{4}{\omega }_{j}^{\frac{2}{3}}$ 时，通过函数 ${h}_{\omega }\left(\delta \right)$ 的单调性证明结论。因为 $y>\frac{3}{4}{\omega }^{\frac{2}{3}}$，且 $\frac{\omega }{8}{\left(\frac{x}{3}\right)}^{-\frac{2}{3}}\in \left(0,1\right)$，则由(2.7)可得 ${\Phi }_{\omega }\left(y\right)\in \left(0,\frac{\pi }{2}\right)$，由此可得函数 $\mathrm{cos}\left(\frac{2}{3}-\frac{2}{3}{\Phi }_{\omega }\left(y\right)\right)$ 单调递减。因此， ${h}_{{\omega }_{i}}\ge {h}_{{\omega }_{j}}$ 成立。

$\frac{2}{3}{y}_{i}\left(1+\mathrm{cos}\left(\frac{2\pi }{3}-\frac{2}{3}\Phi \right)\right)>\frac{2}{3}{y}_{j}\left(1+\mathrm{cos}\left(\frac{2\pi }{3}-\frac{2}{3}\Phi \right)\right),\text{}\forall {y}_{i}>{y}_{j}$.

${h}_{{\omega }_{i}}\left({y}_{i}\right)\ge {h}_{{\omega }_{j}}\left({y}_{j}\right),\text{}\forall {\omega }_{i}<{\omega }_{j},\text{}{y}_{i}>{y}_{j}$.

${Η}_{\lambda }\left(Y\right):=U{\sum }_{{H}_{\lambda }\left(\sigma \right)}{V}^{\text{T}}$ (2.14)

${Η}_{\lambda }\left(Y\right):=\underset{X\in {R}^{m×n}}{\mathrm{arg}\mathrm{min}}\left\{{‖X-Y‖}_{F}^{2}+\lambda {‖X‖}_{\frac{1}{2}}^{\frac{1}{2}}\right\}$ (2.15)

3. 最优算法及其收敛性分析

$\begin{array}{l}{C}_{W}\left(X\right):={‖{P}_{\Omega }\left(X\right)-{P}_{\Omega }\left(Y\right)‖}_{F}^{2}+{‖X‖}_{W,\frac{1}{2}}^{\frac{1}{2}}\\ {C}_{W,\mu }:=\mu \left({C}_{W}\left(X\right)-{‖{P}_{\Omega }\left(X\right)-{P}_{\Omega }\left(Z\right)‖}_{F}^{2}\right)+{‖X-Z‖}_{F}^{2}.\\ {B}_{\mu }\left(Z\right):=Z+\mu \left({P}_{\Omega }\left(Y\right)-{P}_{\Omega }\left({X}^{*}\right)\right)\end{array}$ (3.1)

${B}_{\mu }\left({X}^{*}\right):=\text{}{X}^{*}+\mu \left({P}_{\Omega }\left(Y\right)-{P}_{\Omega }\left({X}^{*}\right)\right)$

${B}_{\mu }\left({X}^{*}\right)$ 的SVD为

${U}^{*}{\sum }_{{B}_{\mu }\left({X}^{*}\right)}{\left({V}^{*}\right)}^{\text{T}}$

${X}^{*}\in {Η}_{W}\left({B}_{\mu }\left({X}^{*}\right)\right)$ (3.2)

${\sigma }_{i}\left({X}^{*}\right)=\left\{\begin{array}{c}{h}_{{\omega }_{i}\mu ,\frac{1}{2}}\left({\sigma }_{i}\left({B}_{\mu }\left({X}^{*}\right)\right)\right),\text{}{\sigma }_{i}\left({B}_{\mu }\left({X}^{*}\right)\right)>\frac{\sqrt[3]{54}}{4}{\left({\omega }_{i}\mu \right)}^{\frac{2}{3}}\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }0,\text{}{\sigma }_{i}\left({B}_{\mu }\left({X}^{*}\right)\right)\le \frac{\sqrt[3]{54}}{4}{\left({\omega }_{i}\mu \right)}^{\frac{2}{3}}\end{array}$ (3.3)

${h}_{{\omega }_{i}}\left(y\right)=\left\{\begin{array}{c}{h}_{{\omega }_{i}\mu ,\frac{1}{2}}\left({\sigma }_{i}\right)，{\sigma }_{i}>\frac{\sqrt[3]{54}}{4}{\left({\omega }_{i}\mu \right)}^{\frac{2}{3}}\\ \text{ }\text{ }\text{ }\text{\hspace{0.17em}}0，{\sigma }_{i}\le \frac{\sqrt[3]{54}}{4}{\left({\omega }_{i}\mu \right)}^{\frac{2}{3}}\end{array}$ (3.4)

${h}_{{\omega }_{i}，\frac{1}{2}}\left({\sigma }_{i}\right)=\frac{2}{3}y\left(1+\mathrm{cos}\left(\frac{2\pi }{3}-\frac{2}{3}{\Phi }_{{\omega }_{i}}\right)\right)$ (3.5)

${\Phi }_{{\omega }_{i}}\left(y\right)=\mathrm{arccos}\left(\frac{{\omega }_{i}\mu }{8}{\left(\frac{y}{\sqrt[3]{54}}\right)}^{-\frac{2}{3}}\right)$ (3.6)

$\begin{array}{c}{C}_{W,\mu }\left(X,Z\right):=\mu \left({C}_{W}\left(X\right)-{‖{P}_{\Omega }\left(X\right)-{P}_{\Omega }\left(Z\right)‖}_{F}^{2}\right)+{‖X-Z‖}_{F}^{2}\\ =‖X‖-2〈X,Z+\mu \left({P}_{\Omega }\left(Y\right)-{P}_{\Omega }\left(Z\right)\right)〉+\mu {‖X‖}_{W,{S}_{\frac{1}{2}}}^{\frac{1}{2}}\\ +‖Z‖+\mu {‖{P}_{\Omega }\left(Y\right)‖}_{F}^{2}-\mu {‖{P}_{\Omega }\left(Z\right)‖}_{F}^{2}\end{array}$

$\begin{array}{c}=‖X‖-2〈X,{B}_{\mu }\left(Z\right)〉+\mu {‖X‖}_{W,{S}_{\frac{1}{2}}}^{\frac{1}{2}}\\ +‖Z‖+\mu {‖{P}_{\Omega }\left(Y\right)‖}_{F}^{2}-\mu {‖{P}_{\Omega }\left(Z\right)‖}_{F}^{2}\\ ={‖X-{B}_{\mu }\left(Z\right)‖}_{F}^{2}+\mu {‖X‖}_{W,{S}_{\frac{1}{2}}}^{\frac{1}{2}}+‖Z‖\\ +\mu {‖{P}_{\Omega }\left(Y\right)‖}_{F}^{2}-\mu {‖{P}_{\Omega }\left(Z\right)‖}_{F}^{2}-{‖{B}_{\mu }\left(Z\right)‖}_{F}^{2}\end{array}$ (3.7)

(3.7)的最后一个等式表明，最小化 ${C}_{W,\mu }\left(X,Z\right)$ 相当于求解问题

$\underset{X\in {R}^{m×n}}{\mathrm{min}}\left\{{‖X-{B}_{\mu }\left(Z\right)‖}_{F}^{2}\text{+}\mu {‖X‖}_{W,{S}_{\frac{1}{2}}}^{\frac{1}{2}}\right\}.$ (3.8)

$\mathrm{min}\left\{{x}_{i}^{2}-2{x}_{i}{\left[{B}_{\mu }\left(Z\right)\right]}_{i}\text{+}\mu {\omega }_{i}{|{x}_{i}|}^{2}\right\}$ (3.9)

${x}_{i}^{2}-{\left[{B}_{\mu }\left(Z\right)\right]}_{i}\text{+}\mu {\omega }_{i}\frac{\text{sign}\left({x}_{i}\right)}{4\sqrt{|{x}_{i}|}}=0$ (3.10)

${\omega }_{i}=\frac{\sqrt{96}}{9{\mu }_{0}}{\left({\left[\sigma \left({X}_{k}\right)\right]}_{{r}_{k}}\right)}^{\frac{3}{2}}$ (3.11)

${r}_{k}$ 表示矩阵 ${X}_{k}$ 的秩。为延续 ${\omega }_{k}$ 的取值，将向量 ${\omega }_{k}$ 更新为

${\omega }_{k+1}=\mathrm{max}\left\{\stackrel{¯}{\lambda },\mathrm{min}\left\{\eta {\omega }_{k,}\frac{\sqrt{96}}{9{\mu }_{0}}{\left({\left[\sigma \left({X}_{k}\right)\right]}_{{r}_{k}}\right)}^{\frac{3}{2}}\right\}\right\}$ (3.12)

(1) 假设 ${X}^{\text{*}}$$\left\{{X}_{k}\right\}$ 的任一聚点，则 ${C}_{\lambda }\left({X}_{k}\right)$ 单调递减收敛于 ${C}_{\lambda }\left({X}^{\text{*}}\right)$

(2) ${X}_{k}$ 是渐进正则的，即 ${\mathrm{lim}}_{k\to \infty }‖{X}_{k+1}-{X}_{k}‖=0$

(3) $\left\{{X}_{k}\right\}$ 的任一聚点均为问题(1.4)的全局最优解。

4. 实验结果与分析

$\frac{{‖{X}_{k+1}-{X}_{k}‖}_{F}}{\mathrm{max}\left\{1,{‖{X}_{k}‖}_{F}\right\}}

$rel:=\frac{{‖M-\stackrel{¯}{X}‖}_{F}}{{‖M‖}_{F}}.$

${M}_{L}\in {R}^{m×r}$${M}_{R}\in {R}^{n×r}$，则 $M={M}_{L}{M}_{R}^{T}$$SR=\frac{p}{mn}$ 为采样比，p是采样数。此外，对于矩阵类型的定义 [19] 如下。一个矩阵称为“简单”的满足： $\frac{p}{r\left(m+n-r\right)}×SR>0.5$$\frac{p}{r\left(m+n-r\right)}>2.6$，“复杂”矩阵定义为 $\frac{p}{r\left(m+n-r\right)}×SR\le 0.5$$\frac{p}{r\left(m+n-r\right)}\le 2.6$

4.1. 相同的尺寸，不同的抽样比和等级

Table 1. Comparison of HFPA and WHFPA for randomly created small but hard matrices (m = n = 100, r = 8:4:20, xtol = 10−6)

4.2. 相同的采样比，不同的维度和等级

Table 2. Comparison of HFPA and WHFPA for randomly created large but hard matrices (SR = 0.570, xtol = 10−4)

4.3. 有噪声的随机矩阵

${B}_{ij}={M}_{ij}+{Z}_{ij}$

Table 3. Comparison of HFPA and WHFPA for randomly created noise disturbance matrices (m = n = 1000, xtol = 10−4)

5. 结论

[1] Li, W., Zhao, L. and Lin, Z. (2015) Non-Local Image Inpainting Using Low-Rank Matrix Completion. Computer Graphics Forum, 34, 111-122.
https://doi.org/10.1111/cgf.12521

[2] Candes, E.J. and Recht, B. (2009) Exact Matrix Completion via Convex Optimization. Foundations of Computational Mathematics, 9, 717-772.
https://doi.org/10.1007/s10208-009-9045-5

[3] Gogna, A. and Majumdar, A. (2015) Matrix Completion Incor-porating Auxiliary Information for Recommender System Design. Expert Systems with Applications, 42, 5789-5799.
https://doi.org/10.1016/j.eswa.2015.04.012

[4] Candes, E.J. and Emmanuel, J. (2010) The Power of Convex Relaxation: Near-Optimal Matrix Completion. IEEE Transactions on Information Theory, 56, 2053-2080.
https://doi.org/10.1109/TIT.2010.2044061

[5] Candes, E.J. and Recht, B. (2009) Exact Matrix Completion via Convex Optimization. Foundations of Computational Mathematics, 9, 717-772.
https://doi.org/10.1007/s10208-009-9045-5

[6] Recht, B., Fazel, M. and Parrilo, P.A. (2010) Guaranteed Mini-mum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization. SIAM Review, 52, 471-501.
https://doi.org/10.1137/070697835

[7] Hu, Y., Zhang, D. and Ye, J. (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

[8] Zhang, D., Hu, Y. and Ye, J. (2013) Matrix Completion by Trun-cated Nuclear Norm Regularization. IEEE Conference on Computer Vision and Pattern Recognition, Vol. 35, 2192-2199.

[9] Candes, E.J. and Plan, Y. (2010) Matrix Completion with Noise. Proceedings of the IEEE, 98, 925-936.
https://doi.org/10.1109/JPROC.2009.2035722

[10] Koltchinskii, V., Lounici, K. and Tsybakov, A.B. (2011) Nu-clear-Norm Penalization and Optimal Rates for Noisy Low-Rank Matrix Completion. The Annals of Statistics, 39, 2302-2329.
https://doi.org/10.1214/11-AOS894

[11] Xu, Z., Chang, X., Xu, F. and Zhang, H. (2012) Regu-larization: A Thresholding Representation Theory and a Fast Solver. IEEE Transactions on Neural Networks and Learning Systems, 23, 1013-1027.
https://doi.org/10.1109/TNNLS.2012.2197412

[12] Gu, S., Zhang, L. and Zuo, W. (2014) Weighted Nuclear Norm Minimization with Application to Image Denoising. 2014 IEEE Conference on Computer Vision and Pattern Recognition, Vol. 1, 2862-2869.
https://doi.org/10.1109/CVPR.2014.366

[13] Oliveira, J.P., Bioucas-Dias, J.M. and Figueiredo, M.A.T. (2009) Adaptive Total Variation Image Deblurring: A Majorization Minimization Approach. Signal Processing, 89, 1683-1693.
https://doi.org/10.1016/j.sigpro.2009.03.018

[14] Lu, Z., Zhang, Y. and Liu, X. (2015) Penalty Decomposition Methods for Rank Minimization. Optimization Methods and Software, 30, 531-558.
https://doi.org/10.1080/10556788.2014.936438

[15] Lai, M., Xu, Y. and Yin, W. (2013) Improved Iteratively Reweighted Least Squares for Unconstrained Smoothed Minimization. SIAM Journal on Numerical Analysis, 51, 927-957.
https://doi.org/10.1137/110840364

[16] Mohan, K. and Fazel, M. (2012) Iterative Reweighted Algo-rithms for Matrix Rank Minimization. Journal of Machine Learning Research, 13, 3441-3473.

[17] Rao, G., Peng, Y. and Xu, Z. (2013) Robust Sparse and Low-Rank Matrix Decomposition Based on the Modeling. Science Chi-na-Information Sciences, 43, 733-748.

[18] Xu, Z.B., Guo, H., Wang, Y. and Zhang, H. (2012) The Representation of Regularizer among Regularizer: An Experimental Study Based on Phase Diagram. Acta Automatica Sinica, 38, 1225-1228.
https://doi.org/10.3724/SP.J.1004.2012.01225

[19] Peng, D., Xiu, N. and Yu, J. (2017) Regularization Methods and Fixed Point Algorithms for Affine Rank Minimization Problems. Computational Optimization and Ap-plications, 67, 543-569.
https://doi.org/10.1007/s10589-017-9898-5

[20] Mirsky, L. (1975) A Trace Inequality of John von Neumann. Monatshefte für Mathematik, 79, 303-306.
https://doi.org/10.1007/BF01647331

[21] Xie, Y., Gu, S. and Liu, Y. (2016) Weighted Schatten p-Norm Mini-mization for Image Denoising and Background Subtraction. IEEE Transactions on Image Processing, 25, 4842-4857.
https://doi.org/10.1109/TIP.2016.2599290

[22] Ma, S., Goldfarb, D. and Chen, L. (2011) Fixed Point and Bregman Iterative Methods for Matrix Rank Minimization. Mathematical Programming, 128, 321-353.
https://doi.org/10.1007/s10107-009-0306-5

[23] Drineas, P., Kannan, R. and Mahoney, M.W. (2006) Fast Monte Carlo Algorithms for Matrices q: Computing Low-Rank Approximations to a Matrix. SIAM Journal on Computing, 36, 158-183.
https://doi.org/10.1137/S0097539704442696

[24] Jain, P., Meka, R. and Dhillon, I. (2010) Guaranteed Rank Minimization via Singular Value Projection. Proceedings of the Advances in Neural Information Processing Systems, Vol. 1, 937-945.

[25] Cai, J.F., Candes, E.J. and Shen, Z.W. (2010) A Singular Value Thresholding Algo-rithm for Matrix Completion. SIAM Journal on Optimization, 20, 1956-1982.
https://doi.org/10.1137/080738970

[26] Xu, C., Lin, Z.C. and Zha, H.B. (2017) A Unified Convex Surrogate for the Schatten-p Norm. Proceedings of the Thirty- First AAAI Conference on Artificial Intelligence, Vol. 25, 926-932.

Top