playground:playground
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| ====== PlayGround ====== | ====== PlayGround ====== | ||
| - | 以下是对课件的补充和扩展： | ||
| - | 2.不适用于非高斯噪声污染的数据集。\\ | ||
| - | 设数据$m\times n$矩阵$M=L+S$，$L$为潜在的低秩矩阵，$S$噪声矩阵。如果$S$非高斯噪声，例如稀疏且幅值不定的噪声，那么PCA将失效。此时，宜用改进的模型RPCA(Robust PCA)。 | ||
| - | RPCA通过以下目标函数求解： | ||
| - | <jsmath>\min_{L,S}\|L\|_{*}+\lambda \|S\|_{l_{1}} \;\; s.t. \; M=L+S</jsmath> | ||
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| - | $\lambda$是权重参数，通常设为$\lambda=\frac{1}{\sqrt{\max\{m,n\}}}$。 $\|\cdot\|_{*}$为核范数，即矩阵奇异值之和。$\|\cdot\|_{l_{1}}$为一范数，即矩阵元素绝对值之和。\\ | ||
| - | 此凸函数具有唯一最小值。使用ALM(Augmented Lagrange Multiplier)求解，最小化增强的拉格朗日函数： | ||
| - | <jsmath>\min_{L,S,Y}l(L,S,Y)=\min_{L,S,Y}\|L\|_{*}+\lambda \|S\|_{l_{1}}+ tr\{Y^{T}(M-L-S)\}+\frac{\mu}{2}\|M-L-S\|_{F}^{2}</jsmath> | ||
| - | $\mu$为另一权重，可取值$\mu=nm/(4\|M\|_{1})$。$\|\cdot\|_{F}$为Frobenius范数，即矩阵元素平方和开根号。\\ | ||
| - | 通过迭代计算求最优值:\\ | ||
| - | 1.初始化：$S_{0}=Y_{0}=0$，$k=0$\\ | ||
| - | 2.while $\|M-L-S\|_{F}>10^{-7}\|M\|_{F}$ do\\ | ||
| - | 3. $L_{k+1}=D_{1/\mu}(M-S_{k}-Y_{k}/\mu)$\\ | ||
| - | 4. $S_{k+1}=S_{\lambda/\mu}(M-L_{k+1}+Y_{k}/\mu)$\\ | ||
| - | 5. $Y_{k+1}=Y_{k}+\mu (M-L_{k+1}-S_{k+1})$\\ | ||
| - | 6. $k=k+1$\\ | ||
| - | 7.end while\\ | ||
| - | 8.输出$L,S$\\ | ||
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| - | 对于标量$x$，函数$S_{a}(x)=sgn(x)max\{|x|-a,0\}$，即截断操作。当输入为矩阵时，对每一元素独立操作。函数$D_a(X)$也是截断操作，但作用于矩阵奇异值：设矩阵$X$的SVD分解为$X=U\Sigma V^{T}$，那么$D_a(X)=US_{a}(\Sigma)V^{T}$。 | ||
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| - | 对RPCA做一些修改，可用于人脸对齐与去噪。这是CVPR2010一篇文章的工作：\\ | ||
| - | RASL: Robust Alignment by Sparse and Low-rank Decomposition for Linearly Correlated Images, Yigang Peng, Arvind Ganesh, John Wright, Wenli Xu, and Yi Ma. To appear in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2010. | ||
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| - | <note important>浙江大学2008-2009版权所有，如需转载或引用，请与[[zhx@cad.zju.edu.cn | 作者联系]]。</note> | ||
playground/playground.1272373762.txt.gz · Last modified: 2023/08/19 21:01 (external edit)
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