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On a notion of averaged operators in CAT(0) spaces

Arian Berdellima

TL;DR

The paper extends the averaged-operator framework from Hilbert spaces to CAT(0) spaces by introducing α-firmly nonexpansive operators through the discrepancy Δ_T. It develops calculus rules showing that compositions and convex combinations of quasi α-firmly nonexpansive operators preserve the class, and proves that iterates of a quasi α-firmly nonexpansive nonexpansive operator converge weakly to a fixed point, with strong projection convergence under regularity. The theory is then instantiated in cyclic and averaged projection schemes, yielding convergence guarantees in complete CAT(0) spaces and connecting to classical Cimmino-type methods. Collectively, these results provide a robust fixed-point and projection framework for nonlinear metric spaces, enabling proximal-like iterative schemes in CAT(0) geometries with explicit convergence properties.

Abstract

Averaged operators have played an important role in fixed point theory in Hilbert spaces. They emerged as a necessity to obtain solutions to fixed point problems where the underlying operator is not contractive and thus renders Banach fixed point theorem inaccessible. We introduce a notion of averaged operator in the broader class of $\text{CAT}(0)$ spaces. We call these operators $α$-firmly nonexpansive and develop basic calculus rules for the quasi $α$-firmly nonexpansive operators. In particular compositions of quasi $α$-firmly nonexpansive operators is quasi $α$-firmly nonexpansive and convex combination of a finite family of quasi $α$-firmly nonexpansive operators is again quasi $α$-firmly nonexpansive. For a nonexpansive operator $T:X\to X$ acting on a $\text{CAT}(0)$ space $X$ we show that the iterates $x_n:=Tx_{n-1}$ converge weakly to some element in the fixed point set $\text{Fix} T$ whenever $T$ is quasi $α$-firmly nonexpansive. Moreover under a certain regularity condition the projections $P_{\text{Fix} T}x_n$ converge strongly to this weak limit. Our theory is illustrated with two classical examples of cyclic and averaged projections.

On a notion of averaged operators in CAT(0) spaces

TL;DR

The paper extends the averaged-operator framework from Hilbert spaces to CAT(0) spaces by introducing α-firmly nonexpansive operators through the discrepancy Δ_T. It develops calculus rules showing that compositions and convex combinations of quasi α-firmly nonexpansive operators preserve the class, and proves that iterates of a quasi α-firmly nonexpansive nonexpansive operator converge weakly to a fixed point, with strong projection convergence under regularity. The theory is then instantiated in cyclic and averaged projection schemes, yielding convergence guarantees in complete CAT(0) spaces and connecting to classical Cimmino-type methods. Collectively, these results provide a robust fixed-point and projection framework for nonlinear metric spaces, enabling proximal-like iterative schemes in CAT(0) geometries with explicit convergence properties.

Abstract

Averaged operators have played an important role in fixed point theory in Hilbert spaces. They emerged as a necessity to obtain solutions to fixed point problems where the underlying operator is not contractive and thus renders Banach fixed point theorem inaccessible. We introduce a notion of averaged operator in the broader class of spaces. We call these operators -firmly nonexpansive and develop basic calculus rules for the quasi -firmly nonexpansive operators. In particular compositions of quasi -firmly nonexpansive operators is quasi -firmly nonexpansive and convex combination of a finite family of quasi -firmly nonexpansive operators is again quasi -firmly nonexpansive. For a nonexpansive operator acting on a space we show that the iterates converge weakly to some element in the fixed point set whenever is quasi -firmly nonexpansive. Moreover under a certain regularity condition the projections converge strongly to this weak limit. Our theory is illustrated with two classical examples of cyclic and averaged projections.

Paper Structure

This paper contains 18 sections, 21 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminary definitions and notations
  3. $\mathop{\mathrm{CAT}}\nolimits(0)$ spaces
  4. Convexity and projections
  5. Nonexpansive operators
  6. Discrepancy of an operator
  7. $\alpha$-firmly nonexpansive operators
  8. Weak convergence
  9. Calculus of quasi $\alpha$-firmly nonexpansive operators
  10. Some preliminary results
  11. Compositions of quasi $\alpha$-firmly nonexpansive operators
  12. Convex combinations
  13. Convex combinations of elements
  14. Convex combinations of operators
  15. Cyclic and averaged projections
  16. ...and 3 more sections

Key Result

Proposition 1

Let $(X,d)$ be a complete $\mathop{\mathrm{CAT}}\nolimits(0)$ space. Let $f: X\to (-\infty,+\infty]$ be a lower semicontinuous strongly convex function with parameter $\mu>0$. Then $f$ has a unique minimizer $x\in X$ and each minimizing sequence converges to $x$. Moreover

Theorems & Definitions (40)

  • Proposition 1: Bacak
  • Definition 1
  • Definition 2
  • Definition 3: $\Delta$-convergence, Lim Lim (1976)
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 30 more