Exact Convergence Rates of Alternating Projections for Nontransversal Intersections

TL;DR

This paper determines the precise convergence rates of the alternating projection method when intersecting a semialgebraic set with a linear subspace, revealing conditions for linear or sublinear convergence based on polynomial multiplicities.

Contribution

It provides explicit formulas for convergence rates using polynomial multiplicities and introduces a method to decide these rates for specific data, extending to higher-dimensional subspaces.

Findings

Exact convergence rates are expressed by polynomial multiplicities.

Conditions for linear vs. sublinear convergence are characterized.

Upper bounds for higher-dimensional intersections are established.

Abstract

We consider the convergence rate of the alternating projection method for the nontransversal intersection of a semialgebraic set and a linear subspace. For such an intersection, the convergence rate is known as sublinear in the worst case. We study the exact convergence rate for a given semialgebraic set and an initial point, and investigate when the convergence rate is linear or sublinear. As a consequence, we show that the exact rates are expressed by multiplicities of the defining polynomials of the semialgebraic set, or related power series in the case that the linear subspace is a line, and we also decide the convergence rate for given data by using elimination theory. Our methods are also applied to give upper bounds for the case that the linear subspace has the dimension more than one. The upper bounds are shown to be tight by obtaining exact convergence rates for a specific semialgebraic set, which depend on the initial points.