0
$\begingroup$

Following is a doubt I got when I was reading Csiszar's Annals of Statistics paper "Why least squares and maximum entropy: An axiomatic approach to inference for linear inverse problems". He comes up with a set of axioms to justify $\ell_2$-norm minimisation in $\mathbb{R}^n$.

He also argues that $\ell_2$-norm minimization would not be appropriate if the space we consider consists of vectors with positive components,say, $\mathbb{R}_+^n$. Further he says that there exists sets of the form

$$L=\{v \in \mathbb{R}_+^n :Av=b\}$$

where $A$ is some $k\times n$ matrix and $u\notin L$ such that $\min_{v\in L}\|v-u\|_2$ is not attained in $L$.

I can't think of such an example.

Can someone think of such an example?

Improve this question
$\endgroup$
2
  • 4
    $\begingroup$ This has nothing to do with algebraic geometry. $\endgroup$
    – Angelo
    Commented Mar 13, 2011 at 5:43
  • $\begingroup$ sorry, I have removed that tag now. $\endgroup$
    – Ashok
    Commented Mar 13, 2011 at 12:14

2 Answers 2

Reset to default
2
$\begingroup$

HI Ashok, I think $\mathbb{R}^n_+$ requires strictly positive components. Then it's easy to come up with the situation above, namely consider $L$ to be the part of the line $x+y = 1$ in $\mathbb{R}_+^2$. And $u = (2,1)$. Then there is no closest point in $L$ to $u$, because it wants to be $(1,0)$, which has nonpositive component. So I guess $n=2$ and $A = (1,1)$ is a $1\times 2$ matrix, and $b=2$.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thanks John Jiang. I didn't think that I would get such an easy example. $\endgroup$
    – Ashok
    Commented Mar 13, 2011 at 12:32
1
$\begingroup$

Let $A=(1 -1)$ and $b=(0)$. Then $L=\left\{{a\choose a}\vert a\in \mathbb R_+\right\}$, so for instance for $u={\ \ 1\choose -1}$ $\min_{v\in L}\|v-u\|_2=\sqrt{2}$ is not attained in $L$.

The point of the example is that the projection of $u$ to the linear space genrated by $L$ is is on the closure of $L$ but not in $L$. In other words, the problem is that $L$ is not closed.

Improve this answer
$\endgroup$
14
  • $\begingroup$ Haha, looks like you are in less need for a point booster. $\endgroup$
    – John Jiang
    Commented Mar 13, 2011 at 4:52
  • 1
    $\begingroup$ @John Jiang: ?? $\endgroup$
    – Georges Elencwajg
    Commented Mar 13, 2011 at 9:46
  • 1
    $\begingroup$ @Sandor and Suvrit: Light-hearted indeed. I have never gotten a single correct answer before so would like to see how many points it's worth:) $\endgroup$
    – John Jiang
    Commented Mar 14, 2011 at 0:18
  • 1
    $\begingroup$ Dear Ashok, there is no reason to be ashamed, the point of this website (I think) is to help each other's understanding of mathematics. We all commit mistakes (I definitely do that a lot!) and overlook things. No harm in that. Take care! $\endgroup$
    – Sándor Kovács
    Commented Mar 14, 2011 at 5:36
  • 1
    $\begingroup$ Let me add, that unfortunately it is really hard to convey one's mood in a comment and one often reads something into what someone else wrote that the other person did not mean. An innocent comment may seem offensive for another person and can lead to bad feelings. I definitely did not mean to do that to you! :) $\endgroup$
    – Sándor Kovács
    Commented Mar 14, 2011 at 5:38

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .