Alternating projections on non-tangential manifolds

TL;DR

This paper proves convergence of alternating projections on intersecting manifolds and shows the limit is close to the optimal point, with applications to frequency estimation in signals.

Contribution

It provides new convergence guarantees for alternating projections on non-tangential manifolds and bounds the limit's proximity to the optimal intersection point.

Findings

Sequences converge under certain conditions.

Limit points are close to the optimal intersection.

Application demonstrated in frequency estimation.

Abstract

We consider sequences $(B_k)_{k=0}^\infty$ of points obtained by projecting back and forth between two manifolds $\M_1$ and $\M_2$, and give conditions guaranteeing that the sequence converge to a limit $B_\infty\in\M_1\cap\M_2$. Our motivation is the study of algorithms based on finding the limit of such sequences, which have proven useful in a number of areas. The intersection is typically a set with desirable properties, but for which there is no efficient method of finding the closest point $B_{opt}$ in $\M_1\cap\M_2$. We prove not only that the sequence of alternating projections converges, but that the limit point is fairly close to $B_{opt}$, in a manner relative to the distance $\|B_0-B_{opt}\|$, thereby significantly improving earlier results in the field. A concrete example with applications to frequency estimation of signals is also presented.