基于加权核范数和<i>L</i><sub>2,1</sub>范数的最优均值线性分类器

曾德宇; 梁泽逍; 吴宗泽

doi:10.11999/JEIT211434

基于加权核范数和L_2,1范数的最优均值线性分类器

doi: 10.11999/JEIT211434

1.
广东工业大学自动化学院广州 510006
2.
广东工业大学粤港澳离散制造智能化联合实验室广州 510006

基金项目: 广东省重点领域研发计划(2021B0101200005)，国家自然科学基金 (62073088, U1911401) , 广东省基础与应用基础研究基金 (2019A1515011606)

详细信息

作者简介:
曾德宇：男，1993年生，博士生，研究方向为模式识别与机器学习

梁泽逍：男，1993年生，博士生，研究方向为多模态信息融合

吴宗泽：男，1975年生，教授，博士生导师，研究方向为智能制造、物联网、人工智能

通讯作者:
吴宗泽　zzwu@gdut.edu.cn

中图分类号: TP273
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出版历程
- 收稿日期: 2021-12-06
- 修回日期: 2022-04-14
- 网络出版日期: 2022-04-21
- 刊出日期: 2022-05-25

Optimal Mean Linear Classifier via Weighted Nuclear Norm and L_2,1 Norm

1.
School of Automation, Guangdong University of Technology, Guangzhou 510006, China
2.
Guangdong-HongKong-Macao Joint Laboratory for Smart Discrete Manufacturing, Guangdong University of Technology, Guangzhou 510006, China

Funds: Guangdong Province Key Field R&D Program (2021B0101200005), The National Natural Science Foundation of China (62073088, U1911401), Guangdong Province Basic and Applied Basic Research Fund (2019A1515011606)

摘要

摘要: 缺陷检测是智能制造系统的一个重要的环节。在采用传统机器学习算法进行缺陷分类的时候，通常会遇到数据噪声干扰，降低算法对缺陷类别的预测精度。尽管近几年提出了如鲁棒线性判别分析(RLDA)等强大的算法用于解决数据受稀疏噪声干扰的分类问题，但仍存在一些缺点限制其应用性能。该文提出一种新的基于线性判别分析的最优均值鲁棒线性分类模型(OMRLSA)。不同于以往应对噪声数据的分类方法忽略稀疏噪声具有的拉普拉斯分布特性对数据均值的影响，该文所提出的最优均值鲁棒线性分类模型会自动更新数据的最优均值，从而保证数据的统计特性不会受到噪声的干扰。此外，随后的损失函数中首次在鲁棒分类模型中引入了关于正则化和误差测量的联合L_2,1范数最小化和秩压缩的加权核范数最小化方法，从而提高算法的鲁棒性。在具有不同比例损坏的标准数据集上的实验结果说明了该文方法的优越性。
- 缺陷检测 /
- 缺陷分类 /
- 鲁棒线性分类器 /
- 低秩恢复
Abstract: Defect detection is an important part of intelligent manufacturing system. When traditional machine learning algorithms are used for defect classification, data noise interference is usually encountered, which reduces the algorithm’s prediction accuracy for defect classification. Although powerful algorithms such as Robust Linear Discriminant Analysis (RLDA) have been proposed in recent years to solve classification problems with data disturbed by sparse noise, there are still some drawbacks that limit its application performance. In this paper, a new Optimal Mean-Robust Linear Classification Analyis (OMRLSA) based on linear discriminant analysis is proposed. Different from the previous classification methods dealing with noisy data, ignoring the influence of the Laplace distribution characteristic of sparse noise on the data mean, the optimal mean robust linear classification model proposed in this paper will automatically update the optimal mean of the data. This ensures that the statistical characteristics of the data will not be disturbed by noise. Furthermore, a weighted kernel norm minimization method with joint L_2,1 norm minimization and rank compression on regularization and error measurement is introduced for the first time in a robust classification model in the subsequent loss function. Thereby the robustness of the algorithm is improved. Experimental results on standard dataset with different ratio corruption illustrate the superiority of the proposed method.
- Defect detection /
- Defect classification /
- Robust linear classification /
- Low rank recovery

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图 1 改进模型

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图 2 YaleB 数据集的一些样本

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图 3 AR数据集的一些样本

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图 4 ORL数据集的一些样本

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图 5 $ \eta $和$ \lambda $的性能

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表 1 基于交替方向乘子法求解问题式(12)

开始:正则化每个样本$ \left\{ {{{\boldsymbol{x}}_i}} \right\} $的2范数为1；
初始化：$ \mu = 1.2 $，$ 1 \lt {\rho} \lt 2 $，$ {\boldsymbol{D}} = {\boldsymbol{X}} $，$ {\boldsymbol{E}} = 0 $，　　　　　${ {\boldsymbol{\varGamma } }_1} = \dfrac{ {\boldsymbol{X} } }{ {\left\\| {\boldsymbol{X} } \right\\|_{\rm{F}}^2} },{ {\boldsymbol{\varGamma } }_2} = { {\boldsymbol{\varGamma } }_1},{\mu _{\max } } = {10^6}$；
(1) 通过求解$\arg {\min _{\boldsymbol{B} } } = \dfrac{ {\eta} }{2}\left\\| { {\boldsymbol{Y} } - { {\boldsymbol{B} }^{\text{T} } }{ {\hat {\boldsymbol{D} } } } } \right\\|_{\rm{F}}^2$$ + \dfrac{{\gamma}}{2}{\left\\| {\boldsymbol{B}} \right\\|_{2,1}} $更新B；
(2) 通过求解$\arg {\min _{ { {\hat {\boldsymbol{D} } } } } } = \dfrac{ {\eta} }{2}\left\\| { {\boldsymbol{Y} } - { { {\boldsymbol{B} }\hat {\boldsymbol{D} } } } } \right\\|_{\rm{F} }^2$ 　　 $+\dfrac{ {\gamma} }{2}\left\\| { { {\hat {\boldsymbol D} } } - \left[ { {\boldsymbol{D} };{ {\bf{1} }^{\text{T} } } } \right] + \dfrac{ { { {\boldsymbol{\varGamma } }_2} } }{\mu } } \right\\|_{\rm{F} }^2$更新$ {{\hat {\boldsymbol D}}} $；
(3) 通过求解$\dfrac{1}{\mu }{\left\\| {\boldsymbol{D} } \right\\|_} + \dfrac{1}{2}\left( {\left\\| { {\boldsymbol{D} } - {\boldsymbol{P} } } \right\\|_{\rm{F}}^2 + \left\\| { {\boldsymbol{D} } - {\boldsymbol{Q} } } \right\\|_{\rm{F} }^2} \right)$更新　　 b,D*，其中　　 ${\boldsymbol{P} } = {\boldsymbol{X} } - {\boldsymbol{E} } - {\boldsymbol{b} }{ {{{\textit{1}}} }^{\text{T} } } + \dfrac{ { { {\boldsymbol{\varGamma } }_1} } }{\mu }$, $ {\boldsymbol{Q}} = {\left[ {{{\hat {\boldsymbol D}}} + \dfrac{{{{\boldsymbol{\varGamma }}_2}}}{\mu }} \right]_{\left( {1:{\text{d}}x,:} \right)}} $；
(4) 通过求解$\dfrac{ {\text{λ } } }{\mu }{\left\\| {\boldsymbol{E} } \right\\|_{2,1} } + \dfrac{1}{2}\left\\| { {\boldsymbol{E} } - { {\hat {\boldsymbol X} } } } \right\\|_{\rm{F}}^2$更新E 　　其中${ {\hat {\boldsymbol X} } } = {\boldsymbol{X} } - {\boldsymbol{E} } - {\boldsymbol{b} }{ {{{\textit{1}}} }^{\text{T} } } + \dfrac{ { { {\boldsymbol{\varGamma } }_1} } }{\mu }$；
(5) 通过${ {\boldsymbol{\varGamma } }_1} = { {\boldsymbol{\varGamma } }_1} + \mu \left( { {\boldsymbol{X} } - {\boldsymbol{D} } - {\boldsymbol{E} } - {\boldsymbol{b} }{ {{{\textit{1}}}}^{\text{T} } } } \right)$ 　　 ${ {\boldsymbol{\varGamma } }_2} = { {\boldsymbol{\varGamma } }_2} + \mu \left( { { {\hat {\boldsymbol D} } } - \left[ { {\boldsymbol{D} };{ {{{\textit{1}}} }^{\text{T} } } } \right]} \right)$和　　 $ \mu = \min \left( {{\rho}\mu ,{\mu _{\max }}} \right) $更新${{\boldsymbol{\varGamma}} }_{1}，{{\boldsymbol{\varGamma}} }_{2}$和$ \mu $；
(6) 判断是否收敛;
输出：B, D, E*

下载: 导出CSV

表 2 单次迭代计算复杂度分析

	加法	乘法	复杂度
计算${\boldsymbol{\varSigma } }$	d_x	d_xc	O(d_xc)
计算B	(d_x+1)²	(d_x+1)²n+(d_x+1)²c	O((d_x+1)²n)
计算${ {\hat {\boldsymbol{D} } } }$	(d_x+1)²	(d_x+1)²c+(d_x+1)²cn	O((d_x+1)²c)
计算b	d_xn	d_xn	O(d_xn)
计算D	d_xn	d_xn²+d_x²n	O(d_xn²)
计算E	n	d_xn	O(d_xn)

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表 3 各个算法在ORL数据集、AR 数据集和YaleB 数据集的识别率

AR数据集	0%	5%	10%	15%	20%	25%	30%	35%	40%
K-NN	46.80±0.94	46.38±0.92	44.75±0.73	44.14±0.76	42.72±0.96	42.22±0.39	41.38±0.58	39.72±0.23	40.22±0.59
LDA	95.91±0.27	95.69±0.26	95.85±0.34	95.06±0.33	94.66±0.53	94.02±0.54	92.09±0.88	91.46±0.79	89.22±0.64
SVM	97.15±0.29	96.89±0.30	96.43±0.34	95.42±0.27	95.45±0.65	95.60±0.53	94.43±0.47	94.03±0.42	92.80±0.41
SRC	79.83±0.59	79.18±0.63	78.48±0.50	78.02±0.43	76.22±0.44	75.54±1.04	75.48±0.96	73.78±0.71	73.80±1.23
RPCA+LDA	96.38±0.36	95.34±0.51	94.02±0.10	91.66±0.77	90.60±1.02	88.69±0.73	86.03±0.46	84.42±1.38	81.88±0.64
RLDA	96.17±0.42	95.95±0.57	95.91±0.53	95.71±0.27	95.68±0.67	95.31±0.41	94.77±0.21	94.68±0.49	93.17±0.44
BDLRR	97.45±0.43	96.22±0.33	96.18±0.42	95.03±0.78	92.81±2.36	90.64±1.77	89.91±1.89	83.24±0.68	80.30±0.27
OMRLDA	97.83±0.29	97.63±0.30	97.62±0.32	97.03±0.14	97.03±0.30	96.35±0.31	95.58±0.43	95.29±0.42	93.45±0.57

ORL数据集	0%	5%	10%	15%	20%	25%	30%	35%	40%
K-NN	86.60±1.75	85.30±2.36	83.60±2.95	82.10±2.82	80.80±2.56	79.00±2.67	78.60±2.63	77.00±2.74	75.00±2.87
LDA	92.40±0.82	90.60±0.96	90.10±1.08	88.30±1.64	87.60±1.43	85.80±2.20	85.20±1.82	83.90±2.04	83.30±2.28
SVM	93.80±1.08	92.90±1.56	92.60±0.66	92.30±1.11	92.10±0.84	92.10±0.75	91.70±0.98	91.17±0.42	90.80±0.41
SRC	93.60±1.64	93.00±1.46	91.90±1.78	91.70±1.10	90.90±1.56	90.10±2.82	89.90±2.30	88.90±1.39	88.80±1.40
RPCA+LDA	92.90±1.19	91.30±1.35	91.40±1.02	89.30±1.48	88.60±1.75	86.60±1.56	86.00±1.90	84.70±2.44	84.50±2.26
RLDA	94.10±1.43	93.00±1.27	92.50±1.58	91.80±1.96	89.60±1.52	88.20±1.30	88.60±1.39	87.10±1.92	87.00±2.15
BDLRR	94.20±2.11	92.90±1.64	93.50±1.73	92.60±1.85	92.10±0.89	90.20±1.04	90.30±1.10	90.60±1.19	89.70±1.04
OMRLDA	94.30±1.64	94.30±1.35	94.00±1.77	92.90±1.75	92.80±1.14	92.50±1.20	92.30±0.91	92.30±0.97	91.60±0.74

YaleB数据集	0%	5%	10%	15%	20%	25%	30%	35%	40%
K-NN	77.90±0.85	77.34±0.52	76.67±0.97	75.58±0.64	74.69±0.16	74.29±0.76	72.17±0.95	70.63±0.63	70.01±0.53
LDA	88.55±1.12	88.54±0.81	87.70±1.77	87.60±1.60	87.41±1.40	87.28±1.89	87.14±0.92	86.88±2.00	85.59±1.18
SVM	95.83±0.69	94.88±0.75	94.42±0.67	94.32±0.70	93.66±0.73	93.09±0.87	92.20±1.13	92.20±0.96	90.69±1.27
SRC	94.52±0.63	93.47±1.10	93.24±0.79	92.89±0.51	92.53±0.93	92.25±0.89	91.97±0.82	91.52±0.78	91.40±1.01
RPCA+LDA	88.85±1.12	88.52±1.03	88.37±1.59	88.74±1.61	87.79±1.47	87.66±1.78	87.18±0.90	86.95±1.96	85.55±1.28
RLDA	97.07±0.42	96.31±0.65	95.66±0.97	94.98±0.73	93.90±1.08	92.76±0.63	91.75±1.08	91.25±1.24	89.65±1.05
BDLRR	96.92±0.65	96.60±0.68	96.14±0.75	95.73±0.58	95.31±0.56	94.68±0.78	94.12±0.74	93.71±0.54	92.48±0.71
OMRLDA	98.31±0.50	97.37±0.69	96.57±0.90	96.12±0.58	95.36±0.91	94.52±0.68	94.33±0.91	94.11±0.82	93.52±0.88

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