The problem is to show that the following iterations converge to the vector $\mu$ and the matrix $\Sigma$.
We have data in the form of nx1 vectors $\mathbf{Q}_k$, $1 \leq k \leq N$ where $$\begin{bmatrix} \mathbf{Q}_k\\ \mathbf{R}_k \end{bmatrix}=U_k \mathbf{X}_k $$ where $U_k$ are known unitary 2nx2n matrices of the form $$ U_k=\begin{bmatrix} \cos(\theta_{1k}) & \sin(\theta_{1k}) & 0 & 0 & 0&\cdots \\ 0 & 0 & \cos(\theta_{2k}) & \sin(\theta_{2k}) & 0&\cdots\\ \vdots & & \vdots & & \vdots\\ -\sin(\theta_{1k}) & \cos(\theta_{1k}) & 0 & 0 &\cdots \\ 0 & 0 & -\sin(\theta_{2k}) & \cos(\theta_{2k}) & 0&\cdots\\ \vdots & & \vdots & & \vdots \end{bmatrix} $$ (permuted rotation matrices). Note that the angles are taken to be independent random values distributed uniformly on $[ 0, 2\pi]$.
$\mathbf{R}_k$ are unknown, and $\mathbf{X}_k \sim \mathcal{N}(\mu,\Sigma)$, where $\mu$ and $\Sigma$ are unknown.
To estimate $\mu$ and $\Sigma$, we iterate as follows:
Let our current estimate for $\mu$ and $\Sigma$ be $m$ and $S$, respectively. Let $$ \begin{bmatrix} m_{Q_k}\\ m_{R_k} \end{bmatrix}=U_km \\ \begin{bmatrix} S_{Q_k Q_k} & S_{Q_k R_k}\\ S_{R_k Q_k} & S_{R_k R_k} \end{bmatrix}=U_kSU_k^T $$ Where the components are equally-sized block-matrices. We then form $$ m_{R_k|Q_k}=m_{R_k}+S_{R_k Q_k}S_{Q_k Q_k}^{-1}\left (Q_k-m_{Q_k} \right ) \\ S_{R_k|Q_k}=S_{R_k R_k}-S_{R_k Q_k}S_{Q_k Q_k}^{-1}S_{Q_k R_k} $$ Then, our updated estimates for $\mu$ and $\Sigma$ are $$ \\ m'=\frac{1}{N}\sum_{k=1}^{N}U_k^T\begin{bmatrix} Q_k\\ m_{R_k|Q_k} \end{bmatrix} \\ S'=\frac{1}{N}\sum_{k=1}^{N}U_k^T\begin{bmatrix} Q_kQ_k^T & Q_km_{R_k|Q_k}^T\\ m_{R_k|Q_k}Q_k^T & S_{R_k|Q_k}+m_{R_k|Q_k}m_{R_k|Q_k}^T \end{bmatrix}U_k -m'm'^T \\ $$ We then iterate.
My goal is to prove that this iteration converges for arbitrary starting vector $m$ and arbitrary positive-definite starting matrix $S$, and, in the limit as $N \rightarrow \infty$, that this converges to $\mu$ and $\Sigma$.
Right now, I'm still having trouble with the n=1 case
Does anyone know of a method to do this? Any help would be greatly appreciated. A bound on some sort of distance measure in terms of starting point and number of iterations would also be great.