$$\alpha $$-Firmly nonexpansive operators on metric spaces

Abstract

We extend to p-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying that the individual components have the required regularity pointwise at fixed points of the splitting algorithms. Our convergence analysis differs from what can be found in the previous literature in that the regularity assumptions are only with respect to fixed points. Indeed we show that, if the fixed point mapping is pointwise nonexpansive at all cluster points, then these cluster points are in fact fixed points, and convergence of the sequence follows. Additionally, we provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates. This allows one for the first time to prove convergence of a tremendous variety of splitting algorithms in spaces with curvature bounded from above.

Convergence of Proximal Splitting Algorithms in CAT(k) Spaces and Beyond

“…(v). This is a specialization of part (iv) when f is the indicator function of a convex set C and follows from the fact that, on symmetric perpendicular p-uniformly convex spaces, the projector is pointwise α-firmly nonexpansive at all points in C with constant α = 1/2 (no violation) as shown in [4,Proposition 25]. As noted in Remark 3, any CAT(κ) space is symmetric perpendicular locally, so by Theorem 21(i) and Lemma 7, in every case Fix T = Ω.…”

“…Proof. This is a minor extension of [4,Theorem 11]. By Lemma 9, it suffices to show (11) at all points y ∈ Fix T 1 ∩ Fix T 0 .…”

“…The regularity of a mapping T : G → G is characterized by the behavior of the images of pairs of points under T . A key tool is what has been called the transport discrepancy in [4]:…”

“…follows immediately from Theorem 25 under the assumption that T satisfies (30) and Ω := argmin f ∩ C = ∅. Compositions of projectors in CAT(κ) spaces has been studied in [4] and [1]. We consider Algorithm 1 when the functions f i := ι C i , the indicator functions of closed convex sets C i ⊂ G, where (G, d) is a complete, symmetric perpendicular p-uniformly convex space with constant c. The p-proximal mapping of the indicator function is the metric projector and so by [4, Proposition 25] these are pointwise α-firmly nonexpansive at all points in ∩ i C i (assuming, of course, that this is nonempty) and by [4,Lemma 10] the cyclic projections mapping…”

Quasi $α$-Firmly Nonexpansive Mappings in Wasserstein Spaces

“…the classical works [15,20,21,24]. In the context of nonlinear CAT(0) spaces (quasi) α-firmly nonexpansive mappings were introduced in [9] and later extended in [10] to more general settings. When d = 1 the Wasserstein-2 space is CAT(0) and the theory about (quasi) α-firmly nonexpansive operators follows from [9].…”