We study the convergence of specific inexact alternating projections for two non-convex sets in a Euclidean space. The $\sigma$-quasioptimal metric projection ($\sigma \geq 1$) of a point $x$ onto a set $A$ consists of points in $A$ the distance to which is at most $\sigma$ times larger than the minimal distance $\mathrm{dist}(x,A)$. We prove that quasioptimal alternating projections, when one or both projections are quasioptimal, converge locally and linearly under the usual regularity assumptions on the two sets and their intersection. The theory is motivated by the successful application of alternating projections to low-rank matrix and tensor approximation. We focus on two problems -- nonnegative low-rank approximation and low-rank approximation in the maximum norm -- and develop fast alternating-projection algorithms for matrices and tensor trains based on cross approximation and acceleration techniques. The numerical experiments confirm that the proposed methods are efficient and suggest that they can be used to regularise various low-rank computational routines.

翻译：我们研究欧几里得空间中两个非凸集特定非精确交替投影的收敛性。点$x$到集合$A$的$\sigma$-拟最优度量投影($\sigma \geq 1$)由$A$中满足到$x$的距离至多为最小距离$\mathrm{dist}(x,A)$的$\sigma$倍的点构成。我们证明，在关于两个集合及其交集的通常正则性假设下，当其中一个或两个投影均为拟最优时，拟最优交替投影局部线性收敛。该理论源于交替投影在低秩矩阵与张量近似中的成功应用。我们聚焦于两个问题——非负低秩近似与最大范数低秩近似，并基于交叉近似与加速技术开发适用于矩阵和张量列的快速交替投影算法。数值实验证实所提方法高效，且表明其可用于正则化各类低秩计算程序。