Weak Jacobian of Proximal Operator
Given a convex function $g(x):mathbb{R}^nrightarrow mathbb{R}$, the proximal operator of $g$ is defined as
$P_g(x)=underset{u}{argmin}quad frac{1}{2}||x-u||_2^2+g(u)$.
Since $g(x)$ is convex, the proximal is a singleton, i.e., there is a unique minimizer $u$. Thus, the proximal operator is a vector function $P_g:mathbb{R}^nrightarrowmathbb{R}^n$.
I'm trying to find an expression for the Jacobian (or weak Jacobian) of $P_g$. In the case where $g$ is differentiable, I believe I can find the Jacobian using the implicit function theorem. However, I'm interested in the general case where $g$ is convex but not differentiable.
As first try, I tried to look on the Moreau Envelope of $g$:
$M_g(x)=underset{u}{min}quad frac{1}{2}||x-u||_2^2+g(u)$.
This function is differentiable and its gradient is given by
$nabla M_g(x) = x-P_g(x)$ .
Therefore, if I could compute the Hessian $H_g$ of $M_g$, I'll get
$H_g=I-J_g$
where $J_g$ is the desired Jacobian.
However, I'm not sure under which conditions the Hessian exists (even in the weak sense) and what the expression for it? Moreover, maybe there is another way to approach this.
optimization convex-analysis convex-optimization weak-derivatives
asked Dec 26 at 15:54
Regev Cohen
262
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Regev Cohen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Given a convex function $g(x):mathbb{R}^nrightarrow mathbb{R}$, the proximal operator of $g$ is defined as
$P_g(x)=underset{u}{argmin}quad frac{1}{2}||x-u||_2^2+g(u)$.
Since $g(x)$ is convex, the proximal is a singleton, i.e., there is a unique minimizer $u$. Thus, the proximal operator is a vector function $P_g:mathbb{R}^nrightarrowmathbb{R}^n$.
I'm trying to find an expression for the Jacobian (or weak Jacobian) of $P_g$. In the case where $g$ is differentiable, I believe I can find the Jacobian using the implicit function theorem. However, I'm interested in the general case where $g$ is convex but not differentiable.
As first try, I tried to look on the Moreau Envelope of $g$:
$M_g(x)=underset{u}{min}quad frac{1}{2}||x-u||_2^2+g(u)$.
This function is differentiable and its gradient is given by
$nabla M_g(x) = x-P_g(x)$ .
Therefore, if I could compute the Hessian $H_g$ of $M_g$, I'll get
$H_g=I-J_g$
where $J_g$ is the desired Jacobian.
However, I'm not sure under which conditions the Hessian exists (even in the weak sense) and what the expression for it? Moreover, maybe there is another way to approach this.
optimization convex-analysis convex-optimization weak-derivatives
asked Dec 26 at 15:54
Regev Cohen
262
New contributor
Regev Cohen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
Given a convex function $g(x):mathbb{R}^nrightarrow mathbb{R}$, the proximal operator of $g$ is defined as
$P_g(x)=underset{u}{argmin}quad frac{1}{2}||x-u||_2^2+g(u)$.
Since $g(x)$ is convex, the proximal is a singleton, i.e., there is a unique minimizer $u$. Thus, the proximal operator is a vector function $P_g:mathbb{R}^nrightarrowmathbb{R}^n$.
I'm trying to find an expression for the Jacobian (or weak Jacobian) of $P_g$. In the case where $g$ is differentiable, I believe I can find the Jacobian using the implicit function theorem. However, I'm interested in the general case where $g$ is convex but not differentiable.
As first try, I tried to look on the Moreau Envelope of $g$:
$M_g(x)=underset{u}{min}quad frac{1}{2}||x-u||_2^2+g(u)$.
This function is differentiable and its gradient is given by
$nabla M_g(x) = x-P_g(x)$ .
Therefore, if I could compute the Hessian $H_g$ of $M_g$, I'll get
$H_g=I-J_g$
where $J_g$ is the desired Jacobian.
However, I'm not sure under which conditions the Hessian exists (even in the weak sense) and what the expression for it? Moreover, maybe there is another way to approach this.
optimization convex-analysis convex-optimization weak-derivatives
asked Dec 26 at 15:54
Regev Cohen
262
New contributor
Regev Cohen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Given a convex function $g(x):mathbb{R}^nrightarrow mathbb{R}$, the proximal operator of $g$ is defined as
$P_g(x)=underset{u}{argmin}quad frac{1}{2}||x-u||_2^2+g(u)$.
Since $g(x)$ is convex, the proximal is a singleton, i.e., there is a unique minimizer $u$. Thus, the proximal operator is a vector function $P_g:mathbb{R}^nrightarrowmathbb{R}^n$.
I'm trying to find an expression for the Jacobian (or weak Jacobian) of $P_g$. In the case where $g$ is differentiable, I believe I can find the Jacobian using the implicit function theorem. However, I'm interested in the general case where $g$ is convex but not differentiable.
As first try, I tried to look on the Moreau Envelope of $g$:
$M_g(x)=underset{u}{min}quad frac{1}{2}||x-u||_2^2+g(u)$.
This function is differentiable and its gradient is given by
$nabla M_g(x) = x-P_g(x)$ .
Therefore, if I could compute the Hessian $H_g$ of $M_g$, I'll get
$H_g=I-J_g$
where $J_g$ is the desired Jacobian.
However, I'm not sure under which conditions the Hessian exists (even in the weak sense) and what the expression for it? Moreover, maybe there is another way to approach this.
optimization convex-analysis convex-optimization weak-derivatives
optimization convex-analysis convex-optimization weak-derivatives
asked Dec 26 at 15:54
Regev Cohen
262
New contributor
Regev Cohen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 26 at 15:54
Regev Cohen
262
New contributor
Regev Cohen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 26 at 15:54
Regev Cohen
262
New contributor
Regev Cohen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 26 at 15:54
Regev Cohen
262
asked Dec 26 at 15:54
Regev Cohen
262
262
New contributor
Regev Cohen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Regev Cohen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Regev Cohen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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