菜鸟的菜地: lp norm

The optimization problem :

$||Ax - b||^{2}_{2} + ||x||^{p}_{p}$

we have

$||Dx||^{p}_{p} = \sum (\sqrt{(x_{i,j+1} - x_{i,j})^{2} + (x_{i+1,j} - x_{i,j})^{2}})^{p}$

and

$\frac{\partial ||Dx||^{p}_{p}}{\partial x_{i,j}} = p/2 [ (x_{i,j+1} - x_{i,j})^{2} + (x_{i+1,j} - x_{i,j})^{2} ]^{ p/2 -1} (-2(x_{i,j+1} - x_{i,j}) - 2(x_{i+1,j} - x_{i,j}) ) + p/2 [ (x_{i,j} - x_{i,j-1})^{2} + (x_{i+1,j-1} - x_{i,j-1})^{2} ]^{ p/2 -1} (2x_{i,j}) + p/2 [ (x_{i+1,j+1} - x_{i-1,j})^{2} + (x_{i,j} - x_{i-1,j})^{2} ]^{ p/2 -1} (2(x_{i,j})$

菜鸟的菜地

2011年5月5日星期四

lp norm

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