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 for super-regular sets with transversal 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的具体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$A$中满足到该点距离不超过最小距离$\mathrm{dist}(x,A)$的$\sigma$倍的点组成。我们证明，当一个或两个投影为拟最优时，对于满足横截相交的超正则集合，拟最优交替投影具有局部线性收敛性。该理论的提出源于交替投影在矩阵与张量低秩近似中的成功应用。我们着重研究两个问题——非负低秩近似与最大范数下的低秩近似——并基于交叉逼近与加速技术，为矩阵和张量列（tensor trains）开发出快速交替投影算法。数值实验证实了所提方法的有效性，并表明其可用于正则化各类低秩计算程序。