Alternating projections on nontangential manifolds
Let \(\mathcal M_1,\mathcal M_2\) be manifolds in a finite-dimensional real Hilbert space \(\mathcal K\), and \(\mathcal M=\mathcal M_1\cap\mathcal M_2\). Denote by \(\pi_1\), \(\pi_2\) and \(\pi\) the corresponding metric projections. Given a point \(B\in \mathcal K\), the sequence \((B_k)\) of alternating projections is defined by \(B_0=B,\, B_{k+1}=\pi_1(B_k)\) if \(k\) is even, and \(B_{k+1}=\pi_2(B_k)\) if \(k\) is odd. A classical result of von Neumann (1949) asserts that if \(\mathcal M_1\), \(\mathcal M_2\) are affine linear manifolds, then \((B_k)\) converges to \(\pi(B)\). The authors study the convergence of this sequence in the case of nonlinear manifolds. The key notion used in the paper is that of nontangential points. A point \(A\in \mathcal M_1\cap\mathcal M_2\) is called nontangential if \(T_{\mathcal M_1}(A)\cap T_{\mathcal M_2}(A)=T_{\mathcal M_1\cap\mathcal M_2}(A)\) (where by \(T_{\mathcal N}(A)\) one denotes the tangent space to a manifold \(\mathcal N\) at a point \(A\in \mathcal N),\) a condition that can be also expressed in terms of the positivity of the angle \(\sigma(A)\) between \(\mathcal M_1,\mathcal M_2\) at \(A\). A point \(A\) is called transversal if \(T_{\mathcal M_1}(A)+ T_{\mathcal M_2}(A)=\mathcal K\). Since any transversal point is nontangential, the results obtained in the present paper extend those obtained by \textit{A. S. Lewis} and \textit{J. Malick} [Math. Oper. Res. 33, No. 1, 216--234 (2008; Zbl 1163.65040)] for transversal points. The main result of the paper (Theorem 5.1) asserts that if \(\mathcal M_1\), \(\mathcal M_2\) are \(C^2\)-manifolds and \(A\in \mathcal M_1\cap\mathcal M_2\) is a nontangential point, then for every \(\varepsilon >0\) and \( 1>c>\sigma(A),\) there exists \(r>0\) such that for any \(B\in\mathcal B(A,r)\) (the open ball) the corresponding sequence \((B_k)\) of alternating projections converges to a point \(B_\infty\in \mathcal M_1\cap\mathcal M_2 ,\; \|B_\infty-\pi(B)\|\leq\varepsilon \|B-\pi(B)\|,\) and \(\|B_\infty- B_k\|\leq\)const\(\cdot c^k \|B-\pi(B)\|\). The second part of the paper is concerned with the case of real algebraic varieties. Numerous examples, illustrative drawings, and a numerical example concerned with the correlation matrix of rank \(k\) that is closest (in the Hilbert-Schmidt norm) to an \(n\times n\) symmetric matrix \(B,\) are also included. The bibliography at the end of the paper counts 55 items.
scientific article; zbMATH DE number 6242066
scientific article