Use this tag in question related to the Proximal Operator / Proximal Mapping. It might also be used in question about Proximal Gradient Method and Alternating Direction Method of Multipliers (ADMM).

Dedicated to the Proximal Operator.

Use this tag in question related to the Proximal Operator / Proximal Mapping. It might also be used in question about Proximal Gradient Method and Alternating Direction Method of Multipliers (ADMM).

Dedicated to the Proximal Operator.

155 questions

votes

Consider a function $f(x)$ with $x \in \mathbb{R}^d$. The proximal operator is defined as $\text{prox}(x) = \text{argmin}_{y} \{ f(y) + \frac{1}{2}\|x - y\|_2^2 \}$. Now, we can similarly define for each direction $i = 1, 2, \dots, d$ a proximal…

sudeep5221

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votes

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…

votes

I was wondering if it was possible to characterize the elements in a subdifferential of a seminorm, $\partial \|\cdot\|$, through means of the proximal mapping. I know of therelation
$$u = Prox_f (v) \Leftrightarrow v - u \in \partial f (u), $$
can…

InspectorPing

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- 1
- 8

votes

Dear Convex Optimization Experts,
My question is related to this post: The Proximal Operator of the $ {L}_{\infty} $ (Infinity Norm), but not really same, I think, as I have a constraint. Apologies if it is obvious to extend the answer.
So, I am…

user550103

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votes

How could one solve the following problem:
$$ \operatorname{Prox}_{\gamma f \left( \cdot \right)} \left( y \right) = \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \gamma {\left\| x \right\|}_{1} \quad \text{subject to} \;…

Royi

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votes

I have two matrices $A$ and $B$ of size $n \times n$. I am trying to find the proximal operator of the below functions i.e. assume one of the matrices constant while finding the $L1$ proximal operator of the other matrix $$ g(B) = |AB|_1 \quad f(A)…

newbie

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votes

The proximal map is defined in the sense of an operator as:
$\text{prox}_{\lambda f}(x) = \arg \min_y f(y) + \frac{1}{2\lambda}\|x-y\|^2$
I don't see why it is called a mapping. I suppose it is because $\arg \min$ might "return" not only one value…

今天春天

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votes

I'm trying to derive the ADMM updates for the $\ell_1$ penalized Huber loss:
$$ \arg\min_x \phi_h \left(y - Ax\right) + \gamma\lVert x \rVert_1 $$
where
$$ \phi_h \left( u \right) =
\begin{cases}
\frac{1}{2}u^2, & \text{if } \mid u \mid \leq 1…

Tom Kealy

- 268
- 2
- 10

votes

How to minimize the follow optimization
$$\begin{array}{ll} \text{minimize} & \| \mathbf{A}\mathbf{x} - \mathbf{b} \|^2\\ \text{subject to} & {\mathbf{x}}^{T} \mathbb{1} = \mathbb{1}\\ & \mathbf{x} \geq 0\end{array}$$
where $\mathbf{x} \in R^{n}$…

jason

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votes

I am trying to find the proximal point of the function $ f \left( x \right) = {\left\| x \right\|}_{2} $ where $ x \in \mathbb{R}^{n} $
The Proximal Operator is defined as:
$$ \operatorname{prox}_{\alpha f} \left( y \right) = \arg \min_{x}…

anktsdmcknsy

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- 18

votes

How to calculate proximal operator of $$F(X,Y) = f(X,Y) + \lambda(||X||^2 + ||Y||^2)$$ where $X$ and $Y$ are matrices. The problem is to minimize the given function with respect to X and Y.Idea is i want to use proximal gradient descent.For that i…

Arun Sharma

- 1
- 1

votes

I am trying to minimize and $L_1$ problem with inequality constraints:
\begin{equation}
||Ax - b||_1 \;\text{subject to}\; x > c
\end{equation}
I believe the way to formulate this with ADMM is:
\begin{equation}
\text{minimize}\; f(x) + g(z)…

kip622

- 131
- 3

votes

This is a question that it is not homework but I would like clear.
I have this Proximal interpretation, that is: the solution of the problem is a fixed point of the following mapping:
$$ x^{\ast} \in \ \ \arg \min_{x \in X_{\text{adm}}} \{\…

Rosa Maria Gtz.

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votes

I'm trying to solve following convex optimization function:
$min_W g(W) + h(W)$, where the $g$ is convex and differentiable and $h$ in convex an non-smooth.
$g(W)=||Y-WX||_F^2$ is square loss function.
Note that,$X,W,Y$ all are matrices.
$h(W)$ is…

user2806363

- 245
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votes

I have a sparse matrix factorization problem, where I want to decompose a matrix $X\in \mathbb{R}^{n\times m}$ to $A\in \mathbb{R}^{n\times p}$ and $B\in \mathbb{R}^{p\times m}$, such that $X\approx AB$.
My matrix $X$ might have outlier values and…

Alt

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