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Generalized Alternating Projections on Manifolds and Convex Sets

Mattias Fält, Pontus Giselsson

Abstract

In this paper, we extend the previous convergence results for the generalized alternating projection method applied to subspaces in [arXiv:1703.10547] to hold also for smooth manifolds. We show that the algorithm locally behaves similarly in the subspace and manifold settings and that the same rates are obtained. We also present convergence rate results for when the algorithm is applied to non-empty, closed, and convex sets. The results are based on a finite identification property that implies that the algorithm after an initial identification phase solves a smooth manifold feasibility problem. Therefore, the rates in this paper hold asymptotically for problems in which this identification property is satisfied. We present a few examples where this is the case and also a counter example for when this is not.

Generalized Alternating Projections on Manifolds and Convex Sets

Abstract

In this paper, we extend the previous convergence results for the generalized alternating projection method applied to subspaces in [arXiv:1703.10547] to hold also for smooth manifolds. We show that the algorithm locally behaves similarly in the subspace and manifold settings and that the same rates are obtained. We also present convergence rate results for when the algorithm is applied to non-empty, closed, and convex sets. The results are based on a finite identification property that implies that the algorithm after an initial identification phase solves a smooth manifold feasibility problem. Therefore, the rates in this paper hold asymptotically for problems in which this identification property is satisfied. We present a few examples where this is the case and also a counter example for when this is not.

Paper Structure

This paper contains 15 sections, 25 theorems, 147 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Subspaces
  5. Manifolds
  6. Generalized Alternating Projections
  7. Manifolds
  8. Convex sets
  9. Examples
  10. Counter example
  11. Conclusions
  12. Appendix
  13. Proof of Theorem \ref{['thm:gapP1-1']}
  14. Lemmas
  15. Proof of Theorem \ref{['thm:gapiff']}

Key Result

Lemma 3.1

If ${\mathcal{M}}$ is a ${\mathcal{C}}^k$ manifold (with $k\geq 2$) around $\bar{x}\in {\mathcal{M}}$, then $\Pi_{{\mathcal{M}}}$ is well defined and ${\mathcal{C}}^{k-1}$ around $\bar{x}$. Moreover $\mathrm{J}_{\Pi_{\mathcal{M}}}(\bar{x})=\Pi_{\mathrm{T}_{{\mathcal{M}}}(\bar{x})}$.

Figures (2)

  • Figure 1: Illustration of manifolds ${\mathcal{M}}$ and ${\mathcal{N}}$ in $\mathbb{R}^2$ and the approximation by tangent planes at a point $\bar{x}\in{\mathcal{M}}\cap{\mathcal{N}}$.
  • Figure 2: Illustration of the problem with a cone $C$ and line $D$ from Example \ref{['ex:counter']}. The iterates $p_0,p_1,p_2,\dots$ are illustrated in red, the normal cone to $C$ with dashed lines, and the rays through $(1,-\gamma)$ and $(-1,-\gamma)$ are shown with blue dotted lines. As shown in the example, the iterates stay on the dotted lines and alternate between projecting on the two faces of $C$.

Theorems & Definitions (55)

  • Definition 3.1: projection
  • Definition 3.2: relaxed projection
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8: smooth manifold
  • Definition 3.9: tangent space and tangent plane
  • Definition 3.10: normal vector
  • ...and 45 more