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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?

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This has nothing to do with algebraic geometry. –  Angelo Mar 13 '11 at 5:43
    
sorry, I have removed that tag now. –  Ashok Mar 13 '11 at 12:14

2 Answers 2

up vote 2 down vote accepted

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 $b=2$.

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Thanks John Jiang. I didn't think that I would get such an easy example. –  Ashok Mar 13 '11 at 12:32

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.

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Haha, looks like you are in less need for a point booster. –  John Jiang Mar 13 '11 at 4:52
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@John Jiang: ?? –  Georges Elencwajg Mar 13 '11 at 9:46
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@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:) –  John Jiang Mar 14 '11 at 0:18
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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! –  Sándor Kovács Mar 14 '11 at 5:36
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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! :) –  Sándor Kovács Mar 14 '11 at 5:38

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