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

The nuclear norm is defined in the following way
$$\|X\|_*=\mathrm{tr} \left(\sqrt{X^T X} \right)$$
I'm trying to take the derivative of the nuclear norm with respect to its argument
$$\frac{\partial \|X\|_*}{\partial X}$$
Note that $\|X\|_*$ is a…

Alt

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votes

I was going through the derivation of soft threholding at http://dl.dropboxusercontent.com/u/22893361/papers/Soft%20Threshold%20Proof.pdf.
It says the three unique solutions for
$\operatorname{arg min} \|x-b\|_2^2 + \lambda\|x\|_1$ is given…

user34790

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votes

What is the proximal operator of the $ \left\| x \right\|_{\infty} $ norm:
$$ \operatorname{Prox}_{\lambda \left\| \cdot \right\|_{\infty}} \left( v \right) = \arg \min_{x} \frac{1}{2} \left\| x - v \right\|_{2}^{2} + \lambda \left\| x…

Alice

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votes

Can you help me explain the basic difference between Interior Point Methods, Active Set Methods, Cutting Plane Methods and Proximal Methods.
What is the best method and why?
What are the pros and cons of each method?
What is the geometric intuition…

Halbort

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votes

Given the prox operator i.e.
$$ \operatorname{prox}_{ h \left( \cdot \right) } \left( x \right) = \arg \min_{u} h \left( u \right) + \frac{1}{2} {\left\| u - x \right\|}_{2}^{2} $$
the Moreau decomposition property says that
$$ x =…

devikad

- 113
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votes

For a convex function $h$, its proximal operator is defined as:
$$\operatorname{prox}_h(x)=\arg\min_u \Big(h(u)+\frac{1}{2}\|u-x\|^2\Big)$$
Can anyone provide an intuitive explanation/motivation of proximal mapping?

mining

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votes

$\newcommand{\prox}{\operatorname{prox}}$
Probably the most remarkable property of the proximal operator is the fixed point property:
The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) $
So, indeed, $f$ can be minimized by find a fixed…

trembik

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votes

Given a convex function $g(x):\mathbb{R}^n\rightarrow \mathbb{R}$, the proximal operator of $g$ is defined as
$P_g(x)=\underset{u}{\arg\min}\quad \frac{1}{2}||x-u||_2^2+g(u)$.
Since $g(x)$ is convex, the proximal is a singleton, i.e., there is a…

Regev Cohen

- 141
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votes

Suppose I have a convex function $f(x)$ for which I can easily compute the proximal mapping
prox$_f(z) = \arg\min_{x} f(x) + \frac{1}{2}||x-z||^2_2$
is there a simple expression for the proximal mapping of $f(\mathcal{A}(x))$ where $\mathcal{A}(x)$…

David P

- 153
- 4

votes

Write down explicitly the optimal solutions to the Moreau-Yosida regularization of the function $f(x)=\lambda\|x\|_1$, where $f:\mathbb{R}^n\to(-\infty,+\infty]$.
I have found that the answer is
$x_i=sgn(x_i)\max\{|x_i|-1,0\}$
Here is my attempt to…

Yellow Skies

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votes

I am encountering an unconstrained minimization problem. The problem is of the form
$$\min_x \frac{\|x-a\|_2^2}{2}+\lambda\|x\|_1$$
where $x,a \in R^n$ and $x$ is the optimization variable. $\lambda \in R$. The problem can be separated in each…

user2104150

- 159
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votes

I was trying to solve
$$\min_x \frac{1}{2} \|x - b\|^2_2 + \lambda_1\|x\|_1 + \lambda_2\|x\|_2,$$
where $ b \in \mathbb{R}^n$ is a fixed vector, and $\lambda_1,\lambda_2$ are fixed scalars. Let $f = \lambda_1\|x\|_1 + \lambda_2\|x\|_2$, that is to…

Luo Zhiheng

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votes

The Douglas-Rachford optimization algorithm solves problems of the form
$$\text{minimize} \hspace{8pt} f(x) + g(x)$$
where $f$ and $g$ are Closed Convex Proper (CCP). It is useful when both $f$ and $g$ have simple proximal operators (in the sense…

NicNic8

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votes

Define the weighted $ {L}_{2} $ norm $ {\left\| x \right\|}_{2,w} = \sqrt{ \sum_{i = 1}^{n} {w}_{i} {x}_{i}^{2} }$. Find the formula for $ \operatorname{prox}_{\lambda
{\left\| \cdot \right\|}_{2,w} }(y)$, where $ \lambda > 0 $.
By definition we…

user112358

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votes

Proximal operator is defined for matrices as a map prox$_f:R^m\times R^n \rightarrow R^m\times R^n$:
prox$_f$(X) := argmin$_{Y\in R^m\times R^n}$ $ f(Y) + \frac{1}{2}||Y-X||^2$
In case of vectors, it is known http://arxiv.org/pdf/0912.3522v4.pdf…

trembik

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