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Given the finite set of matrices $\Bbb X := \{ {\bf X}_1, {\bf X}_2, \dots, {\bf X}_N \} \subset \mathbb{C}^{N_t \times L}$ and the matrix ${\bf Y} \in \mathbb{C}^{N_r \times L}$,

$$ \left( \hat{H}, \hat{X} \right) = \arg \min_{{\bf H} \in \mathbb{C}^{N_r \times N_t}, \\ {\bf X}\in\mathbb{X}} \| {\bf Y} - {\bf H} {\bf X} \|_{\text{F}}^2 $$

where $N_r$ and $N_t$ denote the number of antennas at the receiver and at the transmitter, respectively, and $L$ denotes the total number of time slots.

I am trying to solve the optimization problem above in MATLAB. I do not understand how we proceed. Any help will be highly appreciated.

Rodrigo de Azevedo
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Heretolearn
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1 Answers1

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If the matrix $\bf X$ were fixed and had full row rank, we would have a classical least-squares problem

$$ \hat{{\bf H}} := \arg\min_{{\bf H}} \| {\bf H} {\bf X} - {\bf Y} \|_{\text{F}}^2 = \cdots = {\bf Y} {\bf X}^*\left( {\bf X} {\bf X}^*\right)^{-1}$$

and the minimal least-squares cost

$$ \begin{aligned} \left\| \hat{{\bf H}} {\bf X} - {\bf Y} \right\|_{\text{F}}^2 &= \left\| {\bf Y} {\bf X}^*\left( {\bf X} {\bf X}^*\right)^{-1} {\bf X} - {\bf Y} \right\|_{\text{F}}^2 \\ &= \left\| {\bf Y} \left( {\bf X}^*\left( {\bf X} {\bf X}^*\right)^{-1} {\bf X} - {\bf I} \right) \right\|_{\text{F}}^2 \\ &= \left\| {\bf Y} \left( {\bf I} - {\bf X}^*\left( {\bf X} {\bf X}^*\right)^{-1} {\bf X} \right) \right\|_{\text{F}}^2 \end{aligned} $$

where ${\bf I} - {\bf X}^*\left( {\bf X} {\bf X}^*\right)^{-1} {\bf X}$ is a projection matrix. Since we have $N$ matrices, assuming all of them have full row rank, fix ${\bf X} = {\bf X}_i$, solve the corresponding least-squares problem and obtain the corresponding least-squares cost. At the end, find for which $i$ the least-squares cost is minimal

$$ \hat{i} := \color{blue}{\arg\min_{i \in [N]} \left\| {\bf Y} \left( {\bf I} - {\bf X}_i^*\left( {\bf X}_i {\bf X}_i^*\right)^{-1} {\bf X}_i \right) \right\|_{\text{F}}^2} $$

where $[N] := \{ 1, 2, \dots, N \}$.

Related: Gradient of squared Frobenius norm of a matrix

Rodrigo de Azevedo
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  • @Rodrigo, sir instead of fixing the matrix $\mathbf{X}$, can we fix the matrix $\mathbf{H}$. I am asking this because, I want to plot the Bit error rate which depends on how many bits are received in error for the for the transmitted bits which is $\mathbf{X}$ . – Heretolearn Apr 02 '23 at 14:22
  • @Heretolearn If you fix $\bf H$, then you merely need to try all the ${\bf X}_i$ matrices in order to find out which one yields the minimal least-squares cost – Rodrigo de Azevedo Apr 04 '23 at 21:23
  • Yes go it sir....Thank u so much ....Really learning a lot from highly knowledgeable persons like u... – Heretolearn Apr 05 '23 at 06:17