Find the following transformation $G$

Question

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.

Answer 1

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 $E\{\boldsymbol{x}\boldsymbol{x}^H\} = I$.