Let $(a_j)$ be a sequence with positive terms. There exists $a>0$ such that $a_{j+1}/a_j=1-a/j+r_j$, where $r_j/j\to 0$ as $j\to \infty$.
A question on my book ask me to prove that if $a>1$, then the series $\sum a_j$ converges, but this is clearly not ture. Let $a=2$ and $r_j=\sqrt j$ for $j\g...
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} } $.
What would be the Prox of:
$$ \operatorna...
I would like the proximal operator of the following function:
\begin{align}
f(x) &= \lambda_1 \|x\|_1 + \lambda_2 \|x\|_{2,1} \\
&= \lambda_1 \|x\|_1 + \lambda_2 \sum_g \|x_g\|_2
\end{align}
where $x$ is the concatenation of all $x_g$, $\|x\|_1=\sum_i|x_i|$, and $\|y\|_2=\sqrt{\sum_i y_i...
I want to solve the following problem:
$$ \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 Proximal operator to Huber function ) . However, i...
In mathematical optimization, the proximal operator is an operator associated with a proper, lower-semicontinuous convex function
f
{\displaystyle f}
from a Hilbert space
X
{\displaystyle {\mathcal {X}}}
to
]
−
∞
,
+
∞
]
{\displaystyle ]-\infty ,+\infty ]}
, and is defined by:
prox
f
...
I am reading the ADMM paper by S. Boyd et al - Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers.
I'm interested in implementing a L1 regularized feature wise distributed multinomial logistic regression solver using ADMM. So I read page 70, se...
I created a new tag - proximal-operator. I also creates 2 equivalent (Synonymous) tags - prox-operator and proximal-mapping. Could some approve their Wiki and link them as I intended? Thank You.
I created a new tag - proximal-operator. I also creates 2 equivalent (Synonymous) tags - prox-operator and proximal-mapping. Could some approve their Wiki and link them as I intended? Thank You.
I created a new tag - `proximal-operators`. I also creates 2 equivalent (Synonymous) tags - `prox-operator` and `proximal-mapping`. Could some approve their Wiki and link them as I intended? Thank You. Also, Could someone rename the tag from `proximal-operators` to `proximal-operator`? Thank You.
@Royi I suppose that it would be reasonable to stop adding tag to the questions. Bumping 35 questions in less than hour is more than recommended limit for bumping old questions.
My personal recommendation would be to make an answer in the Tag management thread - to see what other people think about the new tag.
3
@Royi Well, creating the tag synonym seems a bit redundant; as we do not know whether the tag will stay or whether it will be removed.
Anyway, if you want somebody to create a tag synonym, it can be done by moderators or by users who have score +5 (or higher) in the tag.
If you make the post on meta about the tag which you propose, you can include your recommendation for the synonym into that post - so that the people who actually can create the synonym can see it there.
@Royi You need score +5 in the master tag. It is explained here.
> Users with more than 2500 reputation and a total answer score of 5 or more on the tag, can suggest tag synonyms. Users with a total answer score (total upvotes minus total downvotes) of 5 or more on the tag, can vote for tag synonyms. Suggestions will be automatically approved when they reach a score of 4, and automatically deleted when they reach a score of -2.
You will still need other users to vote on the synonym.
Which is why I think it's easiest if it gets cleaned up by the moderators. (Who are not restricted by the score in the tag and who can apply change immediately.)
Certainly, the situation with three distinct tags that were intended to be synonymous is not optimal.
@Royi If somebody suggests a synonym, then the place where users can upvote or downvoted the synonym will be displayed.
You can see this, for example, in ring-theory - as somebody suggested (ring-homomorphism) as a synonym of that tag.
Let $(a_j)$ be a sequence with positive terms. There exists $a>0$ such that $a_{j+1}/a_j=1-a/j+r_j$, where $r_j/j\to 0$ as $j\to \infty$.
A question on my book ask me to prove that if $a>1$, then the series $\sum a_j$ converges, but this is clearly not ture. Let $a=2$ and $r_j=\sqrt j$ for $j\g...
I am reading the ADMM paper by S. Boyd et al - Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers.
I'm interested in implementing a L1 regularized feature wise distributed multinomial logistic regression solver using ADMM. So I read page 70, se...
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 have:
$$ \operatorname{prox}_{\lambda {\left...
