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

I read that for small $\lambda$, the proximal operator will satisfy
$$\mathbf{prox}_{\lambda f}(x)=x-\lambda\nabla f(x)+o(\lambda)$$
How to prove it?
Here
$$\mathbf{prox}_{\lambda f}(x) = \arg \min\limits_y f(y)+\frac{1}{2\lambda}\|x-y\|^2_2$$

1024

- 525
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votes

I want to solve the following problem:
$$ \arg\min_x |f(x)|_\mu + \frac{1}{2\sigma} |x-x^k|^2 $$
, where
$$|x|_\mu = \begin{cases} \frac{|x|^2}{2\mu}, & |x|<\mu \\ |x|-\frac \mu 2 & |x|\geq \mu \end{cases}. $$
If $f(x)=x$, I can solve it ( See…

jakeoung

- 1,221
- 11
- 27

votes

I'm supposed implementing certain optimization algorithms (ISTA, FISTA) to minimize: $$\frac12 ||Ax-(Ax_0+z)||_2^2 + \lambda ||x||_1.$$
$A$ is a matrix, $x$ is a vector, $z$ is some noise filled with random data from a certain distribution.…

Leo 254

- 343
- 1
- 18

votes

If the proximal operator of $f(x)$ is $\text{prox}_{\lambda f}(x)$, what about $cf(x)$ and $f(cx)$, c is a scalar.
For example, If $f(x) = ||x||_{1}$, $x \in \mathbb{R}^{n}$, how about the proximal operator of $c||x||_{1}$ and $||c x||_{1}$.

Xia

- 111
- 12

votes

I was following Gabriel Peyre - Dictionary Learning for Denoising tutorial, In section 21 it is given Proximal operator over a ball $B_\epsilon$ of radius $\epsilon$ as
$$\text{Proj}_{B_\epsilon(y)}(u) = y + (u-y) \max({1 ,…

user52705

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