I use a sampling matrix $\mathbf{\Phi }\in \mathbb{R}^{M\times N} (M\ll N)$ to obtain a measurement vector $\mathbf{y }\in \mathbb{R}^{M}$ from the original signal $\mathbf{x }\in \mathbb{R}^{N}$ by the linear projection of $\mathbf{y }=\mathbf{\Phi}\mathbf{x}$. For further reconstructing the orginal signal $\mathbf{x }$ from the observed $\mathbf{y}$, I use the iterative shrinkage-thresholding algorithm (ISTA) to perform the recovery process by iterating between the following two steps:
$$ \mathbf{r}^{\left( k \right)}=\mathbf{\hat{x}}^{\left( k-1 \right)}-\rho ^{(k)} \mathbf{\Phi }^{\top}\left( \mathbf{\Phi \hat{x}}^{\left( k-1 \right)}-\mathbf{y} \right) ,\\ \mathbf{\hat{x}}^{\left( k \right)}=\underset{\mathbf{\hat{x}}}{\text{arg}\min}\frac{1}{2}\lVert \mathbf{\hat{x}}-\mathbf{r}^{\left( k \right)} \rVert _{2}^{2}+\lambda \lVert \mathbf{\Psi \hat{x}} \rVert _1, $$
where $k$ is the iteration index, $\rho ^{(k)}$ is the step size of gradient descent, $\mathbf{\Psi}$ is related to a pre-defined sparse domain (e.g. DCT), and $\lambda$ is the regularization coefficient.
For initialization, I use $\hat{\mathbf{x}}^{(0)}=\mathbf{\Phi}^\top\mathbf{y}$ to solve the dimensionality mismatch between the measurement vector $\mathbf{y}$ and the signal $\mathbf{x}$. To take a deeper insight into the first step, I simplified the difference between $\mathbf{r}^{(k)}$ and the orginal $\mathbf{x}$, denoted by $\mathbf{d}^{(k)}$, as follows:
$$ \mathbf{d}^{(k)}=\mathbf{r}^{\left( k \right)}-\mathbf{x}=\left( \mathbf{I}-\rho ^{\left( k \right)}\mathbf{\Phi }^{\top}\mathbf{\Phi } \right) \left( \mathbf{\hat{x}}^{\left( k-1 \right)}-\mathbf{x} \right), $$
where $\mathbf{I}\in \mathbb{R}^{N\times N}$ is the identity matrix.
I am confused on how to analyse the statistical feature or some other properties about the elements in the above $\mathbf{d}^{(k)}$. For example, if the sampling matrix $\mathbf{\Phi}$ is a given fixed random Gaussian matrix (all elements are independent and obey the normal distribution $\mathcal{N}(0, 1)$), what about if the Gaussian matrix is orthogonalized with rows? Or $\mathbf{\Phi}$ is a orthogonal binary matrix (all elements belongs to $\left\{ 0,1 \right\}$)?
I am a beginner and know little about matrix analysis methods, so after trying, I decided to ask for help. I am waiting for some analysis and suggestions or tips. Thank you for reading my long question :) !