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

What would be the the Sub Gradient of
$$ f \left( X \right) = {\left\| A X \right\|}_{2, 1} $$
Where $ X \in \mathbb{R}^{m \times n} $, $ {A} \in \mathbb{R}^{k \times m} $ and $ {\left\| Y \right\|}_{2, 1} = \sum_{j} \sqrt{ \sum_{i} {Y}_{i,j}^{2} }…

Royi

- 7,459
- 3
- 40
- 82

votes

\begin{equation}
\arg\min_{X} \frac{1}{2}\|X-Y\|_{F}^2 + \tau\|X\|_{*}
\end{equation}
where $\tau\geq 0,Y\in \mathbb{C}^{n\times n}$ and $\|\cdot\|_{*}$ is the nuclear norm. What's the solution of this convex optimization?
In some literature,…

Chenfl

- 71
- 1
- 5

votes

The problem is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\|x \right\|}_{2} $$
Where $y$ and $x$ are vectors. $\|\cdot\|_2$ is Euclidean norm. In the paper Convex Sparse Matrix Factorizations, they say…

E.J.

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- 1
- 8
- 21

votes

I am working with a non-Euclidean proximity operator defined by a Bregman distance function $D(\cdot, \cdot)$:
$$
\operatorname{prox}_f(x) = \operatorname*{argmin}_u \{ f(u) + D(u, x) \}
$$
Is anybody aware of a result similar to Moreau…

Alex Shtof

- 267
- 6
- 20

votes

Is there an efficient way to evaluate the proximal operator of the function $f:\mathbb R^n \to \mathbb R \cup \{ \infty \}$ defined by
\begin{equation}
f(x) = \| x \|_2 + I_{\geq 0}(x),
\end{equation}
where $I_{\geq 0}$ is the indicator function of…

littleO

- 48,104
- 8
- 84
- 154

votes

I'm trying to come up with a super simple proof of convergence on the proximal point algorithm, which uses the iteration scheme
$x^{i+1} = \mathbf{prox}_{tf}(x^i)$
where $f$ is a closed, convex (but not necessarily smooth) function and…

Y. S.

- 1,634
- 9
- 15

votes

How to compute the proximal mapping (prox-operator) of $f(x)=||Ax||_{2}$?
Here $A$ is a diagonal matrix with all positive eigenvalues.
I know how to compute the prox mapping of $f(x)=||x||_{2}$, but I have not found any connection between these two…

YuzheChen

- 179
- 7

votes

Consider a proximal operator,
$$ \operatorname{Prox}_{ \lambda f( u ) } \left( x \right) = \arg \min_{u} \lambda f \left( u \right) + \frac{1}{2} {\left\| u - \mu x \right\|}_{2}^{2}.$$
What is the partial derivative of the proximal operator w.r.t.…

w382903

- 135
- 9

votes

Let $f\colon H\to\mathbb{R}\cup \{+\infty\}$ is convex, lower semicontinuous, proper and coercive function. $H$ is a Hilbert space.
$f_{\lambda}:H\to\mathbb{R}\cup \{+\infty\}$ is the Moreau-Yosida approximation with $\lambda>0$:…

rrrto2005

- 141
- 1
- 8

votes

Let $f:\mathbb R^n \to \mathbb R$ be a piecewise linear function, i.e.,
$$
f(x) = \max_i \langle a_i, x\rangle + b_i,
$$
for some $i \geq 1$. I'm interested in efficiently solving the proximal operator for small values of $n$.
$$
\text{prox}_{\tau…

yon

- 383
- 1
- 10

votes

How can I calculate the proximal operator of mixed norm $ {L}_{\infty,1} $ for any general matrix, $X\in R^{m\times n}$ i.e.,
$$ {X}^{\ast} = \arg \min_X {\left\| X \right\|}_{\infty, 1} + \frac{1}{2 \tau} {\left\| X - Y \right\|}_{F}^{2} $$
Where $…

Sohil Shah

- 81
- 4

votes

I would like to calculate the proximal operator of spectral norm for any general matrix, $X \in \mathbb R^{m\times n}$, i.e.,
$$X^* = \arg \min_X \|X\|_2 + \frac{1}{2\tau} \|X-Y\|_F^2$$
I understand that the proximal operator for nuclear norm…

Sohil Shah

- 81
- 4

votes

This is an assignment problem which I failed to solve in a couple of days.
Denote the set of all $n \times n$ symmetric matrices and the set of all $n \times n$ symmetric positive definite matrices by $\mathbb{S}^n$ and $\mathbb{S}^n_{++}$…

Empiricist

- 7,628
- 1
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- 40

votes

Is there a closed-form solution of the following convex problem:
$$\min_x \| x - u \| + C \| x - v \|^2$$
where $\| \cdot \|$ is the L2 norm.

user171375

votes

$\newcommand{\prox}{\operatorname{prox}}$
$\newcommand{\argmin}{\operatorname{argmin}}$
$\newcommand{\dom}{\operatorname{dom}}$
Recall again that the proximal operator for vectors $\prox_{f}: R^n \rightarrow R^n$ of $f$ is defined as:
$\prox_f(v) :=…

trembik

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