# Mixed Alternating Projections with Application to Hankel Low-Rank Approximation
## 4 Linearized Cadzow Iterations in the Signal-Estimation Problem
### Limit Property of the Composition of Projections  
First, consider the relation connecting the (semi-)norm \(\|Z\|_{\mathbf{W}}\) generated by the inner product (2) and the (semi-)norm \(\|Z\|_{\mathbf{L},\mathbf{R}}\), which is generated by the inner product (4) (semi-)norms arise if not to require nondegeneracy of the matrices \(\mathbf{W}\), \(\mathbf{L}\) and \(\mathbf{R}\)). The mapping \(\mathcal{T}_{L}\) preserves the norm if \(\|Z\|_{\mathbf{W}}=\|Z\|_{\mathbf{L},\mathbf{R}}\) for any \(Z\), where \(\mathbf{Z}=\mathcal{T}_{L}(\mathbb{Z})\), \(\mathbf{L}\) and \(\mathbf{R}\) are matrix weights and \(\mathbf{W}\) is a weight matrix. Define the convolution of the two matrices \(\mathbf{C}=\mathbf{A}\ast\mathbf{B}\) for \(\mathbf{A}\in\mathbb{R}^{L\times L}\), \(\mathbf{B}\in\mathbb{R}^{K\times K}\) and \(\mathbf{C}\in\mathbb{R}^{N\times N}\) as  
\[c_{i,j}=\sum_{i_{1}+i_{2}-1=i_{1}+j_{2}-1=j}a_{i_{1},i_{2}}b_{i_{2}j_{2}},\]  
where \(\mathbf{A}=(a_{i_{1},i_{1}})\), \(\mathbf{B}=(b_{i_{2},i_{2}})\) and \(\mathbf{C}=(c_{i,j})\). There is a known relation linking the weight matrix to the matrix weights. **Theorem 4** ([8]). : \(\|Z\|_{\mathbf{W}}=\|Z\|_{\mathbf{L},\mathbf{R}}\) _for any series \(\mathbb{Z}\in\mathbb{R}^{N}\) if and only if \(\mathbf{W}=\mathbf{W}(\mathbf{L},\mathbf{R})=\mathbf{L}\ast\mathbf{R}\)._  
All statements of the following theorem are a direct application of Theorem 1 and Proposition 1 with consideration to  
\[\mathbf{P}_{\mathbf{S}_{0},(\mathbf{L},\mathbf{R})}^{\text{tr}}=\Big{(} \mathcal{T}_{L}^{-1}\Pi_{\mathcal{H},(\mathbf{L},\mathbf{R})}\Pi_{\mathcal{N} (\mathbf{S}_{0}),(\mathbf{L},\mathbf{R})}\mathcal{T}_{L}\Big{)}^{n}=\mathcal{T }_{L}^{-1}\Big{(}\Pi_{\mathcal{H},(\mathbf{L},\mathbf{R})}\Pi_{\mathcal{N}( \mathbf{S}_{0}),(\mathbf{L},\mathbf{R})}\Big{)}^{n}\mathcal{T}_{L}\]  
and \(\mathcal{T}_{L}^{-1}(\mathcal{H}\cap\mathcal{N}(\mathbf{S}_{0}))=\mathcal{Z  
_All the eigenvalues of the matrix_ \(\mathbf{P}_{\mathbf{S}_{0},(\mathbf{L},\mathbf{R})}\) _lie within the unit circle; the eigenvalue_ \(\lambda=1\)_, which only has modulus_ \(1\)_, is semi-simple and has multiplicity equal to_ \(2r\)_; the corresponding eigenvectors form the basis of_ \(\mathcal{Z}(\mathbf{a}_{0}^{2})\)_. If the weight matrix_ \(\mathbf{W}\) _is nondegenerate,_ \[\|(\mathbf{P}_{\mathbf{S}_{0},(\mathbf{L},\mathbf{R})}^{\mathsf{H}}-\mathbf{ \Pi}_{\mathcal{Z}(\mathbf{a}_{0}^{2}),\mathbf{L}\ast\mathbf{R}})\chi\|_{ \mathbf{W}}\leq C\|\chi\|_{\mathbf{W}}(|\hat{\lambda}|+\delta)^{n},\] (19) _where_ \(C\) _is some positive constant,_ \(\delta>0\) _is any small real number,_ \(\hat{\lambda}\) _is the largest non-unitary (that is,_ \((2r+1)\)_-th) eigenvalue of the matrix_ \(\mathbf{P}_{\mathbf{S}_{0},(\mathbf{L},\mathbf{R})}\)_._  
Influences of the Matrices \(\mathbf{L}\) and \(\mathbf{R}\) on the First Principal Angle and Convergence Rate of the Cadzow Method  
Recall that if one wants to solve the problem (1) for a given weight matrix \(\mathbf{W}\) using the Cadzow method, one must require \(\mathbf{W}=\mathbf{L}\ast\mathbf{R}\). To obtain the maximum likelihood estimate, the matrix \(\mathbf{W}\) is chosen as the inverse of the noise autocovariance matrix. However, for such weight matrices \(\mathbf{W}\) in the case of white or autoregressive noise, there are no nondegenerate matrix weights \(\mathbf{L}\) and \(\mathbf{R}\). Therefore, the matrix weights \(\mathbf{L}\) and \(\mathbf{R}\) are sought numerically under the upper bound \(1/\alpha\) of the condition number of \(\mathbf{R}\) for a given \(\mathbf{L}\). The works [13; 14] are devoted to the search for appropriate matrix weights. To find these weights, the matrix \(\mathbf{L}\) was fixed in some specific way, and the diagonal matrix \(\mathbf{R}\) was numerically found such that the measure of discrepancy considered in these papers was minimal: \(\rho(\mathbf{W},\mathbf{L}\ast\mathbf{R})\to\min_{\mathbf{R}}\) provided \(\text{cond}\ \mathbf{R}\leq 1/\alpha\), where \(0<\alpha\leq 1\).