# Mixed Alternating Projections with Application to Hankel Low-Rank Approximation
## 3 Mixed Alternating Projections  
Suppose that different norms may be considered in the problem (7). For example, such a case arises if a weighted least-squares (WLS) problem is solved. Such a problem can be solved both with an optimal weight matrix in terms of the minimum error of WLS estimates and with some suboptimal ones. Thus, consider two equivalent norms and assume that the application of the method of alternating projections in the first norm leads to a more accurate estimation of the solution, but the algorithm converges slowly, and alternating projections in the second norm, on the contrary, converge faster, but the accuracy of the solution is worse. We propose a way to speed up the convergence of the algorithm using the first norm, almost without worsening the accuracy of the solution. Let us consider the projectors by two equivalent norms and denote the corresponding matrices \(\mathbf{P}_{\mathcal{A},\mathcal{B}(1)}\) and \(\mathbf{P}_{\mathcal{A},\mathcal{B}(2)}\). This notation is introduced and implies both projections in one of the 1 st or 2 nd norms in the definition \(\mathbf{P}_{\mathcal{A},\mathcal{B}(i)}=\mathbf{\Pi}_{\mathcal{A}(i)}\mathbf{ \Pi}_{\mathcal{B}(i)}\), \(i=1,2\). The mixed alternating projections are defined as:  
\[X_{T}=\mathbf{P}_{\mathcal{A},\mathcal{B}(2)}^{T_{2}}\mathbf{P}_{\mathcal{A}, \mathcal{B}(1)}^{T_{1}}X_{0}, \tag{11}\]  
where \(T=T_{1}+T_{2}\). Since this formula uses only linear operators, we can prove the main theorem of this paper. **Theorem 2**. : _For the algorithm given by (11), the following relations are true, where convergence is in the operator norm:_  
1. \(\mathbf{P}_{\mathcal{A},\mathcal{B}(2)}^{T_{2}}\mathbf{P}_{\mathcal{A}, \mathcal{B}(1)}^{T_{1}}\rightarrow\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}(1)}\) _if_ \(T_{2}\) _is fixed and_ \(T_{1}\rightarrow+\infty\)_._
2. _There exists an equivalent third norm such that_ \(\mathbf{P}_{\mathcal{A},\mathcal{B}(2)}^{T_{2}}\mathbf{P}_{\mathcal{A}, \mathcal{B}(1)}^{T_{1}}\rightarrow\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}(3)}\) _if_ \(T_{1}\) _is fixed and_ \(T_{2}\rightarrow+\infty\)_._  
Proof. : The proof of the first statement follows from Theorem 1: \(\mathbf{P}_{\mathcal{A},\mathcal{B}(1)}^{T_{1}}\rightarrow\mathbf{\Pi}_{ \mathcal{A}\cap\mathcal{B}(1)}\) as \(T_{1}\rightarrow+\infty\) and also \(\mathbf{P}_{\mathcal{A},\mathcal{B}(2)}^{T_{2}}\mathbf{\Pi}_{\mathcal{A}\cap \mathcal{B}(1)}=\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}(1)}\) for any \(T_{2}\). To prove the second statement, we first apply Theorem 1 to the matrix \(\mathbf{P}_{\mathcal{A},\mathcal{B}(2)}^{T_{2}}\) as \(T_{2}\rightarrow+\infty\), and then consider the matrix \(\widetilde{\mathbf{P}}=\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}(2)}\mathbf{P}_ {\mathcal{A},\mathcal{B}(1)}^{T_{1}}\). It is sufficient to show that the matrix \(\widetilde{\mathbf{P}}\) defines a projection--i.e., \(\widetilde{\mathbf{P}}=\widetilde{\mathbf{P}}\widetilde{\mathbf{P}}\); then, the required projection is \(\widetilde{\mathbf{P}}=\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}(3)}\) in a third norm. By using direct substitution, we obtain:  
\[\widetilde{\mathbf{P}}\widetilde{\mathbf{P}}=\mathbf{\Pi}_{\mathcal{A}\cap \mathcal{B}(2)}\biggl{(}\mathbf{P}_{\mathcal{A},\mathcal{B}(1)}^{T_{1}} \mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}(2)}\biggr{)}\mathbf{P}_{\mathcal{A},\mathcal{B}(1)}^{T_{1}}.\]  
Since \(\mathbf{P}_{\mathcal{A},\mathcal{B}(1)}^{T_{1}}\mathbf{\Pi}_{\mathcal{A}\cap \mathcal{B}(2)}=\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}(2)}\) for any \(T_{1}\), the second statement is proved. Theorem 2 has the following interpretation: if one first makes \(T_{1}\) steps with projections in the first norm in the iteration (11), and then iterates with projections in the second norm, then the obtained result will still be a projection on the desired subspace (item 2), and the larger \(T_{1}\) is, the closer this projection will be to the projection in the first norm (item 1).