3
$\begingroup$

I asked this question 6 days ago on math.stackexchange.com (https://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here.

I'm not sure if I'm writing this problem properly, so any suggestions or modifications are much appreciated, but here is what I have:

Let $\mathbf{y=Gx}$, where $\mathbf{G} \in \mathbb{C}^{N\times N}$ and $\mathbf{x}\in \mathbb{C}^{N\times 1}$ is a random vector. Also let $\mathbf{A} \in \mathbb{C}^{M\times N}, M<N$ be a known matrix. Define the error between the original vector $\mathbf{x}$ and the resulting vector $\mathbf{y}$ as: $$\mathbf{\epsilon=x-y=(I-G)x}=[\epsilon_1,\epsilon_2,...,\epsilon_N]^T$$The data in $\mathbf{x}$ are independent and identically distributed, i.e, $E\{\mathbf{x^Hx}\}=\mathbf{I}$. I want to find G that minimizes $$\displaystyle\min_{\mathbf{G}} \|\mathbf{AG}\|_F^2 $$ subject to two constraints:

(i) an average mean square error $E{\mathbf{\|\epsilon\|_2^2}}=\gamma$. I can write this constraint as $\mathbf{\|(I-G)\|_F^2}=\gamma$

(ii) a uniform or constant average error for every entry in $\mathbf{\epsilon}$, i.e, $E{\|\epsilon_1\|^2_2}=E{\|\epsilon_2\|^2_2}=...=E{\|\epsilon_N\|^2_2}$. I don't know how to properly write this constraint in terms of $\mathbf{G}$, so any help?

I'd rather find a way to write the second constraint in terms of $\mathbf{G}$ and solve the problem as it is, but if not then one direction I'm thinking of going to is to design $\mathbf{G}$ as block diagonal, i.e: $$\mathbf{G=diag(G_1,G_2,...G_{k})}$$ where $k=\frac{N}{L}$ and $\mathbf{G_1,...,G_{k}}$ are all $L\times L$ matrices, so instead of a constant average error for every entry I will get a constant average error per block of data and its kinda an approximation to the original problem depending on the size of the block. The problem becomes: $$\displaystyle\min_{\mathbf{G}} \|\mathbf{AG}\|_F^2 \quad{} \text{subject to} \quad{} \|\mathbf{(I-G)}\|_F^2=\gamma, \text{and} ~\|\mathbf{(I-G_1)}\|_F^2=\|\mathbf{(I-G_2)}\|_F^2=...=\|\mathbf{(I-G_k)}\|_F^2=\frac{\gamma}{k},~\gamma \geq0$$

Any advise on whether this is solvable or a better way of writing the second constraint.

$\endgroup$
  • 1
    $\begingroup$ Math.SE is definitely the right place to ask this. I think the lack of an answer simply reflects that it is not an easy problem. $\endgroup$ – Michael Grant Feb 24 '14 at 15:48
  • $\begingroup$ And the title should be changed into a more useful one (the current one provides essentially no information about the topic) $\endgroup$ – YCor Mar 23 '16 at 14:09
1
$\begingroup$

To simplify the statement, I will use the following notation for real vectors and matrices. Let $\epsilon_i = \boldsymbol{e}_i^T \boldsymbol{\epsilon}$, where $\boldsymbol{e}_i \in \mathbb{R}^N$. Sometimes $\boldsymbol{e}_i$ is called the $i$-th standard basis vector; it has all elements equal to 0 except the $i$-th element equal to 1. We have $$ E(\epsilon_i^2) = E[(\boldsymbol{e}_i^T (I - G) \boldsymbol{x})^2] = \boldsymbol{e}_i^T (I - G)(I - G^T) \boldsymbol{e}_i$$ or $$ E(\epsilon_i^2) = \|(I - G^T) \boldsymbol{e}_i\|_2^2. $$ By the way, the given condition $E\{\boldsymbol{x}^H\boldsymbol{x}\} = I$ should be $E\{\boldsymbol{x}\boldsymbol{x}^H\} = I$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.