The Proximity Operator of a Function with Multiple Affine MappingProximal mapping for composition of...
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The Proximity Operator of a Function with Multiple Affine Mapping
Proximal mapping for composition of functionsProximal functionsProximal mapping of $f(U) = -log det(U)$Proof of convergence for the proximal point algorithmWhen is a mapping the proximity operator of some convex function?Does the linear transformation that a matrix encodes depend on a choice of basis?How to derive the proximal operator of the Euclidian norm?How to Do Backtracking Line Search for Proximal Gradient Decent?Low-rank matrix satisfying linear constraints linear mappingProximal Mappings intuition and practical example
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Let $f(mathbf{x}) = g(mathbf{A}mathbf{x})$, where $mathbf{A} in mathbb{R}^{M times N}$ is a linear transformation satisfying $mathbf{A}mathbf{A}^T = mathbf{I}$. Then for any $mathbf{x} in mathbb{R}^{N}$,
begin{equation}
text{prox}_f (mathbf{x}) = mathbf{x} + mathbf{A}^T (text{prox}_g(mathbf{Ax}) − mathbf{Ax}).
end{equation}
Now, if $f(mathbf{x}) = sum_{p=1}^{P} g(mathbf{A}_pmathbf{x})$, where $mathbf{A}_p in mathbb{R}^{M times N}$ are multiple linear trasformations satisfying $mathbf{A}_pmathbf{A}_p^T = mathbf{I}$. Then for any $mathbf{x} in mathbb{R}^{N}$, what would be the proximal mapping for the new $f(mathbf{x})$?.
linear-algebra optimization convex-analysis
asked Mar 18 at 14:07
Héctor Miguel Vargas GarcíaHéctor Miguel Vargas García
112
$endgroup$
add a comment |
$begingroup$
Let $f(mathbf{x}) = g(mathbf{A}mathbf{x})$, where $mathbf{A} in mathbb{R}^{M times N}$ is a linear transformation satisfying $mathbf{A}mathbf{A}^T = mathbf{I}$. Then for any $mathbf{x} in mathbb{R}^{N}$,
begin{equation}
text{prox}_f (mathbf{x}) = mathbf{x} + mathbf{A}^T (text{prox}_g(mathbf{Ax}) − mathbf{Ax}).
end{equation}
Now, if $f(mathbf{x}) = sum_{p=1}^{P} g(mathbf{A}_pmathbf{x})$, where $mathbf{A}_p in mathbb{R}^{M times N}$ are multiple linear trasformations satisfying $mathbf{A}_pmathbf{A}_p^T = mathbf{I}$. Then for any $mathbf{x} in mathbb{R}^{N}$, what would be the proximal mapping for the new $f(mathbf{x})$?.
linear-algebra optimization convex-analysis
asked Mar 18 at 14:07
Héctor Miguel Vargas GarcíaHéctor Miguel Vargas García
112
$endgroup$
add a comment |
$begingroup$
Let $f(mathbf{x}) = g(mathbf{A}mathbf{x})$, where $mathbf{A} in mathbb{R}^{M times N}$ is a linear transformation satisfying $mathbf{A}mathbf{A}^T = mathbf{I}$. Then for any $mathbf{x} in mathbb{R}^{N}$,
begin{equation}
text{prox}_f (mathbf{x}) = mathbf{x} + mathbf{A}^T (text{prox}_g(mathbf{Ax}) − mathbf{Ax}).
end{equation}
Now, if $f(mathbf{x}) = sum_{p=1}^{P} g(mathbf{A}_pmathbf{x})$, where $mathbf{A}_p in mathbb{R}^{M times N}$ are multiple linear trasformations satisfying $mathbf{A}_pmathbf{A}_p^T = mathbf{I}$. Then for any $mathbf{x} in mathbb{R}^{N}$, what would be the proximal mapping for the new $f(mathbf{x})$?.
linear-algebra optimization convex-analysis
asked Mar 18 at 14:07
Héctor Miguel Vargas GarcíaHéctor Miguel Vargas García
112
$endgroup$
Let $f(mathbf{x}) = g(mathbf{A}mathbf{x})$, where $mathbf{A} in mathbb{R}^{M times N}$ is a linear transformation satisfying $mathbf{A}mathbf{A}^T = mathbf{I}$. Then for any $mathbf{x} in mathbb{R}^{N}$,
begin{equation}
text{prox}_f (mathbf{x}) = mathbf{x} + mathbf{A}^T (text{prox}_g(mathbf{Ax}) − mathbf{Ax}).
end{equation}
Now, if $f(mathbf{x}) = sum_{p=1}^{P} g(mathbf{A}_pmathbf{x})$, where $mathbf{A}_p in mathbb{R}^{M times N}$ are multiple linear trasformations satisfying $mathbf{A}_pmathbf{A}_p^T = mathbf{I}$. Then for any $mathbf{x} in mathbb{R}^{N}$, what would be the proximal mapping for the new $f(mathbf{x})$?.
linear-algebra optimization convex-analysis
linear-algebra optimization convex-analysis
asked Mar 18 at 14:07
Héctor Miguel Vargas GarcíaHéctor Miguel Vargas García
112
asked Mar 18 at 14:07
Héctor Miguel Vargas GarcíaHéctor Miguel Vargas García
112
asked Mar 18 at 14:07
Héctor Miguel Vargas GarcíaHéctor Miguel Vargas García
112
asked Mar 18 at 14:07
Héctor Miguel Vargas GarcíaHéctor Miguel Vargas García
112
asked Mar 18 at 14:07
Héctor Miguel Vargas GarcíaHéctor Miguel Vargas García
112
112
add a comment |
add a comment |
1 Answer
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Even in the simpler case where $P=2$ and $A_1=A_2= textbf{I}$, there does not exist a closed form solution for the proximal operator of the sum. If you are interested in solving an optimization problem check the keywords "Douglas-Rachford" and "splitting".
answered 2 days ago
xelxel
1338
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
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$begingroup$
Even in the simpler case where $P=2$ and $A_1=A_2= textbf{I}$, there does not exist a closed form solution for the proximal operator of the sum. If you are interested in solving an optimization problem check the keywords "Douglas-Rachford" and "splitting".
answered 2 days ago
xelxel
1338
$endgroup$
add a comment |
$begingroup$
Even in the simpler case where $P=2$ and $A_1=A_2= textbf{I}$, there does not exist a closed form solution for the proximal operator of the sum. If you are interested in solving an optimization problem check the keywords "Douglas-Rachford" and "splitting".
answered 2 days ago
xelxel
1338
$endgroup$
add a comment |
$begingroup$
Even in the simpler case where $P=2$ and $A_1=A_2= textbf{I}$, there does not exist a closed form solution for the proximal operator of the sum. If you are interested in solving an optimization problem check the keywords "Douglas-Rachford" and "splitting".
answered 2 days ago
xelxel
1338
$endgroup$
Even in the simpler case where $P=2$ and $A_1=A_2= textbf{I}$, there does not exist a closed form solution for the proximal operator of the sum. If you are interested in solving an optimization problem check the keywords "Douglas-Rachford" and "splitting".
answered 2 days ago
xelxel
1338
answered 2 days ago
xelxel
1338
answered 2 days ago
xelxel
1338
answered 2 days ago
xelxel
1338
1338
add a comment |
add a comment |
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