# Mixed Alternating Projections with Application to Hankel Low-Rank Approximation
## 1 Introduction  
**Problem. ** Consider the problem of extracting a signal \(\mathsf{S}=(s_{1},\ldots,s_{N})^{\mathsf{T}}\) from a noisy time series \(\mathsf{X}=\mathsf{S}+\mathsf{N}\), where \(\mathsf{S}\) is governed by a linear recurrence relation (LRR) of order \(r\): \(s_{n}=\sum_{i=1}^{r}a_{i}s_{n-i}\), \(n=r+1,\ldots,N\), \(a_{r}\
eq 0\), \(\mathsf{N}\) is noise with zero expectation and autocovariance matrix \(\mathsf{\Sigma}=\mathsf{\Sigma}_{N}\in\mathbb{R}^{N\times N}\). There is a well-known result that describes the set of time series governed by LRRs in parametric form: \(s_{n}=\sum_{i}P_{i}(n)\exp(\beta_{i}n)\cos(2\pi\alpha_{i}n+\psi_{i})\), where \(P_{i}(n)\) are polynomials in \(n\) (see, e.g., ([1], Theorem 3.1.1)). Methods based on the estimation of signal subspace (subspace-based methods), and in particular, singular spectrum analysis (see [2, 3, 4]), work well for the analysis of such signals. The subspace-based approach starts from fixing a window length \(L\), \(1<L<N\), putting \(K=N-L+1\) and constructing the \(L\)-trajectory matrix for a series \(\mathsf{S}\in\mathbb{R}^{N}\):  
\[\mathsf{S}=\mathcal{T}_{L}(\mathsf{S})= \begin{pmatrix}s_{1}&s_{2}&\ldots&s_{ K}\\ s_{2}&s_{3}&\ldots&s_{K+1}\\ \vdots&\vdots&\vdots&\vdots\\ s_{L}&s_{L+1}&\ldots&s_{N}\end{pmatrix} ,\]  
where \(\mathcal{T}_{L}\) denotes the bijection between \(\mathbb{R}^{N}\) and \(\mathcal{H}\); \(\mathcal{H}\) is the set of Hankel matrices of size \(L\times K\) with the equal values on the auxiliary diagonals \(i+j=\mathrm{const}\). We say that the time series \(\mathsf{S}\) is called a time series of rank \(r\), if its trajectory matrix \(\mathsf{S}\) is rank-deficient and has rank \(r\) for any \(L\), \(r<L<N-r+1\). The equivalent definition of series of rank \(r\), \(2r<N\), is that \((r+1)\)-trajectory matrix has rank \(r\) ([5], Corollary 5.1). Denote the set of time series of rank \(r\) and length \(N\) as \(\mathcal{D}_{r}^{N}=\mathcal{D}_{r}\). Consider the problem of approximating a time series by series of rank \(r\) in a weighted Euclidean norm as the following non-linear weighted least-squares problem:\[\mathsf{Y}^{\star}=\operatorname*{arg\,min}_{\mathsf{Y}\in\overline{\mathcal{D}_{r}}} \|\mathsf{X}-\mathsf{Y}\|_{\mathbf{W}}^{2}, \tag{1}\]  
where \(\mathsf{X},\mathsf{Y}\in\mathbb{R}^{N}\) are time series of length \(N\) and \(\|\mathsf{Z}\|_{\mathbf{W}}\) is the weighted Euclidean norm generated by the inner product  
\[\langle\mathsf{Z}_{1},\mathsf{Z}_{2}\rangle_{\mathbf{W}}=\mathsf{Z}_{1}^{ \mathsf{T}}\mathbf{W}\mathsf{Z}_{2}, \tag{2}\]  
\(\mathbf{W}\in\mathbb{R}^{N\times N}\) is a symmetric positive-definite weight matrix and \(\overline{\mathcal{D}}_{r}\) is the closure of the set \(\mathcal{D}_{r}\). The solution \(\mathsf{Y}^{\star}\) of the problem (1) can be used as a finite-rank estimate of the signal \(\mathsf{S}\). Moreover, the obtained estimate allows one to solve many related problems: estimating the LRR coefficients and time series parameters, forecasting the time series, filling in the gaps (if any) and so on. Note that when \(\mathbf{W}=\mathbf{\Sigma}^{-1}\), the resulting signal estimate \(\mathsf{Y}^{\star}\) assuming Gaussian noise \(\mathsf{N}\) is the maximum likelihood estimate. **Known solution.