# Mixed Alternating Projections with Application to Hankel Low-Rank Approximation
## 1 Introduction
** To solve the problem (1), consider the related problem of structured low-rank approximation (SLRA; see a review in [6]):  
\[\mathbf{Y}^{\star}=\operatorname*{arg\,min}_{\mathbf{Y}\in\mathcal{M}_{r} \cap\mathcal{H}}f_{\mathbf{L},\mathbf{R}}(\mathbf{Y}),\;\;f_{\mathbf{L}, \mathbf{R}}(\mathbf{Y})=\|\mathbf{X}-\mathbf{Y}\|_{\mathbf{L},\mathbf{R}}^{2}, \tag{3}\]  
where \(\|\mathbf{Z}\|_{\mathbf{L},\mathbf{R}}\) is the matrix norm generated by the inner product described in [7]:  
\[\langle\mathbf{Z}_{1},\mathbf{Z}_{2}\rangle_{\mathbf{L},\mathbf{R}}=\operatorname {tr}(\mathbf{L}\mathbf{Z}_{1}\mathbf{R}\mathbf{Z}_{2}^{\mathsf{T}}), \tag{4}\]  
\(\mathcal{H}=\mathcal{H}_{L,K}\) is the space of Hankel matrices of size \(L\times K\); \(\mathcal{M}_{r}\subset\mathbb{R}^{L\times K}\) is the set of matrices of rank not larger than \(r\); \(\mathbf{L}\in\mathbb{R}^{L\times L}\) and \(\mathbf{R}\in\mathbb{R}^{K\times K}\) are positive-definite matrices; and \(\mathbf{X}\) and \(\mathbf{Y}\) are the matrices related to (1) as \(\mathbf{X}=\mathcal{T}_{L}(\mathsf{X})\) and \(\mathbf{Y}=\mathcal{T}_{L}(\mathsf{Y})\). Then, the estimate of the signal \(\mathsf{S}\) is \(\mathcal{T}_{L}^{-1}(\mathbf{Y}^{\star})\). There exists a relation between the weight matrix \(\mathbf{W}\) and the matrices \(\mathbf{L}\) and \(\mathbf{R}\) that provides the equivalence of the problems (1) and (3) [8] and thereby the coincidence of the corresponding signal estimates. The algorithm for the numerical solution of the Hankel SLRA problem (3) is the Cadzow iterative method (introduced in [9]; see also [4]), which belongs to the class of alternating projection methods [10]. Let us define the following sequence of matrices starting from a given matrix \(\mathbf{X}\):  
\[\mathbf{X}_{0}=\mathbf{X},\quad\mathbf{X}_{k+1}=\Pi_{\mathcal{H}_{L}(\mathbf{ L},\mathbf{R})}\Pi_{\mathcal{M}_{r}(\mathbf{L},\mathbf{R})}\mathbf{X}_{k}, \quad k\geq 0, \tag{5}\]  
where \(\Pi_{\mathcal{H}_{L}(\mathbf{L},\mathbf{R})}\) and \(\Pi_{\mathcal{M}_{r}(\mathbf{L},\mathbf{R})}\) are projectors on the corresponding sets in the matrix norm \(\|\cdot\|_{\mathbf{L},\mathbf{R}}\) generated by the inner product (4). As a numerical solution we can take the value at \(T\)-th iteration, that is, \(\mathcal{T}_{L}^{-1}(\mathbf{X}_{T})\), which can then be used as the initial value in local search methods, for example, in VPGN [11] and MGN [12] methods. The problem of search of matrices \(\mathbf{L}\) and \(\mathbf{R}\) (let us call them \"matrix weights\" to differ from the weight matrix \(\mathbf{W}\)) that would correspond to the matrix \(\mathbf{W}_{0}=\mathbf{\Sigma}^{-1}\) is already known. The choice \(\mathbf{W}_{0}=\mathbf{\Sigma}^{-1}\) is natural for a model with noise autocovariance matrix \(\mathbf{\Sigma}\) and leads to the maximum likelihood estimation of the signal. For the case of autoregressive noise (and for white noise in particular), it is shown in [8] that the equivalence of the problems (1) and (3) is possible, only if either the matrix \(\mathbf{L}\) or \(\mathbf{R}\) is degenerate; this is inappropriate when solving the problem with methods of the Cadzow type; see ([13], Remark 4). Thereby, nondegenerate matrices \(\mathbf{L}\) and \(\mathbf{R}\), which give a matrix \(\mathbf{W}\) approximated by \(\mathbf{\Sigma}^{-1}\), have to be used. The papers [8,14] are devoted to solving the problems of finding such matrices. The problem to be solved is parameterized by a parameter \(0<\alpha\leq 1\). Assume that the matrix \(\mathbf{L}\) is fixed and the matrix \(\mathbf{R}\) is subject to the following constraints: \(\mathbf{R}\) is diagonal, and its condition number is bounded--say, \(\operatorname{cond}\mathbf{R}\leq 1/\alpha\)--that is, \(\alpha\) is the parameter regulating conditionality. By decreasing \(\alpha\), we can achieve weights closer to the required weights of \(\mathbf{W}_{0}\), but this reduces the convergence rate of the Cadzow method (5) [13].