In my research in applied probability and signal processing, I have recently come across the following problem involving (mean square error) MSE which I need to compute:
We assume we have a random vector representing a signal, denoted by $ s $, and we assume an additive white noise, which is a random vector $ n $ of the same dimension as $ s $ and statistically independent of $ s $ and by white noise we mean the variance matrix of $ n $ is a scalar matrix $ p_n I $ where $ I $ is the identity matrix. We define the noisy signal as the following random vector $ x = s + n $ we may assume for convenience all random vectors are centered (have zero mean). We wish to solve the following minimization problem, known as the Generalized Karhunen-Loeve Transform:
$ \begin{equation*} \begin{aligned} & \underset{T}{\text{minimize}} & & E\lVert {s-Tx} \rVert ^2 \\ & \text{subject to} & & rank(T)<m \end{aligned} \end{equation*} $
Where the norm is the $ L_2 $ norm. We know the solution to this general problem to be $ T^* = trun_m\{{\Sigma_{sx}\Sigma_x ^ {-\frac{1}{2}}}\} \Sigma_x ^ {-\frac{1}{2}}$ where $ \Sigma $ denotes variance or covariance matrix, and $ trun_m $ denotes $ m $ order SVD truncation of a matrix, meaning if the SVD of $ A $ is $ A = U\Sigma V^* $ where the singular values on the diagonal are in non-increasing order, then we define $ trun_m(A) = \sum_{i=1}^{m} {\sigma_i u_i v_i ^ *} $. In the case of the specific noisy signal problem we have, after defining the spectral decomposition $ \Sigma_x = \sum_{i=1}^{n} \lambda_i e_i e_i^* $ we may obtain that $ T^* = \sum_{i=1}^{m} \frac{\lambda_i - p_n}{\lambda_i} e_i e_i ^* $ but what troubles me is that the actual mean square error $ E\lVert {s-Tx} \rVert ^2 \\$ seems intractable, this error (the expectation) intrigues me as I wish to find it as an expression of $ p_n $ to study the different behaviors for different values $ p_n $, and yet it I cannot find a way to compute it analytically though I think an analytic solution should exist.
That is my problem statement and I cannot find a way to begin to compute the mean square error analytically, numerically I expect it should be OK but I would like to find an analytical solution. I appreciate all help on this.