Convergence of proximal splitting algorithms in CAT(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{CAT}(\kappa)$\end{document} spaces and beyond

In the setting of CAT(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{CAT}(\kappa)$\end{document} spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky–Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric subregularity. Linear metric subregularity is in any case necessary for linearly convergent fixed point sequences, so the result is tight. To show this, we develop a theory of fixed point mappings that violate the usual assumptions of nonexpansiveness and firm nonexpansiveness in p-uniformly convex spaces.

Definition 2 Let (G, d) be a geodesic space and γ and η be two geodesics through p. Then γ is said to be perpendicular to η at point p denoted by γ ⊥ p η if d(x, p) ≤ d(x, y) ∀x ∈ γ , y ∈ η.
A space is said to be symmetric perpendicular if for all geodesics γ and η with common point p we have κ is symmetric perpendicular [9,Theorem 2.11].
In a complete p-uniformly convex space the p-proximal mapping of a proper and lower semicontinuous function f is defined by The main dividend of this work is the following. A more precise statement of this theorem, with proof, is Theorem 25. The intervening sections prove the fundamental building blocks.

Almost α-firmly nonexpansive mappings
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]: (i) The mapping T : G → G is pointwise almost nonexpansive at y ∈ D ⊂ G on D with violation ≥ 0 whenever The smallest for which (4) holds is called the violation. If (4) holds with = 0, then T is pointwise nonexpansive at y ∈ D ⊂ G on D. If (4) holds at all y ∈ D, then T is said to be (almost) nonexpansive on D. If D = G, then the mapping T is simply said to be (almost) nonexpansive. If D ⊃ Fix T = ∅ and (4) holds at all y ∈ Fix T with the same violation, then T is said to be almost quasi nonexpansive. (ii) The mapping T is said to be pointwise asymptotically nonexpansive at y whenever where D (y) is a neighborhood of y in D.
If (7) holds with = 0, then T is pointwise α-firmly nonexpansive at y ∈ D ⊂ G on D. If (7) holds at all y ∈ D with the same constant α, then T is said to be (almost) α-firmly nonexpansive on D. If D = G, then the mapping T is simply said to be (almost) α-firmly nonexpansive. If D ⊃ Fix T = ∅ and (7) holds at all y ∈ Fix T with the same constant α, then T is said to be almost quasi α-firmly nonexpansive. (v) The mapping T is said to be pointwise asymptotically α-firmly nonexpansive at y with constant α < 1 whenever where D (y) is a neighborhood of y in D.
Proposition 6 (Characterizations) Let (G, d) be a p-uniformly convex space with constant c > 0, and let T : For fixed y ∈ Fix T, the function ψ In particular, T is almost quasi α-firmly nonexpansive on D whenever T possesses fixed points and (10) holds at all y ∈ Fix T with the same constant α ∈ [0, 1) and violation at most . Proof This is a slight extension of [4,Proposition 4], which was for pointwise α-firmly nonexpansive mappings. The proof for pointwise almost α-firmly nonexpansive mappings is the same.

Composition of operators
Before continuing with pointwise almost α-firmly nonexpansive mappings, we make a brief but important observation about fixed points of compositions of quasi strictly nonexpansive mappings (6).

Lemma 7 Let T 1 and T 2 be quasi strictly nonexpansive on
Remark 8 The sufficiency of strict quasi nonexpansivity for the analogous identity for convex combinations of mappings in a Hadamard space was recognized in [3, Remark 7.11].

Lemma 9
Let (G, d) be a p-uniformly convex space with constant c > 0, and let D ⊂ G. Let T 0 : D → G be pointwise almost α-firmly nonexpansive at y on D with constant α 0 and violation 0 , and let T 1 : T 0 (D) → G be pointwise almost α-firmly nonexpansive at T 0 y on T 0 (D) with constant α 1 and violation 1 . Then the composition T := T 1 • T 0 is pointwise almost α-firmly nonexpansive at y with constant α ∈ (0, 1) and violation at most = 0 + 1 + 0 1 on D whenever Proof The proof is a slight extension of the same result for compositions of α-firmly nonexpansive mappings in [4, Lemma 10]. Since T 1 is pointwise α-firmly nonexpansive at T 0 y with violation 1 and constant α 1 on T 0 (D), we have where ψ (p,c) for all x ∈ D. Whenever (11) holds, we conclude that where = 0 + 1 + 0 1 .

Averages of operators
Let B(G) be the Borel algebra on (G, d), P be the family of probability measures on (G, B(G)), and P p (G) be the family of probability measures on G such that the pth moment exists i.e.
For μ ∈ P p (G), the minimizer of We use the notation T i x * η for the push forward of η with respect to the mapping i → T i x for fixed x i.e.
Theorem 11 Let G be a proper, symmetric perpendicular, p-uniformly convex space with constant c > 0, and let T i , i ∈ I be a family of almost quasi α-firmly nonexpansive operators with violation i and constant α i respectively on G. Let η be a probability measure on I such that T i x * η ∈ P p (G) for all x ∈ G. Then T = b p (T i , η) is a pointwise almost α-firm operator at any y ∈ i∈I Fix T i on G with constant α = sup i∈I α i and violation at most = sup i∈I i .
Proof Let x ∈ G be arbitrary, y ∈ i∈I Fix T i ⊂ Fix T , ν = T i x * η as defined in (16) and (d(·, y) p ) * ν the push forward of ν. Then And again Jensen's inequality completes the proof For a finite index set I, without loss of generality I = {1, . . . , n} and a probability measure η on I, we can define ω i := η(i) for all i. Then In case of a Hilbert space G and p = c = 2 this reduces further to b p (T i , η)(x) = n i=1 ω i T i x. If the support of the measure ν consists of two discrete points x 1 and x 2 i.e. for ω ∈ [0, 1], ν = ωδ x 1 + (1ω)δ x 2 , then b p (ν) can be calculated explicitly. It is obvious that b p (ν) has to lie on the geodesic connecting x 1 and The next result shows that the convex combination of an almost nonexpansive mapping with the identity mapping can be made arbitrarily close to α-firmly nonexpansive (no violation) by choosing the averaging constant small enough-this can be interpreted as choosing an appropriately small step size. . Then T β := βT ⊕ (1β) Id is pointwise almost α-firmly nonexpansive at all y ∈ Fix T with constant α β = αβ p-1 αβ p-1αβ + 1 and violation at most β := β.

Proposition 12 (Krasnoselsky-Mann relaxations) Let (G, d) be a p-uniformly convex space and T : G → G be pointwise almost nonexpansive at all y ∈ Fix T with violation
and solving for α β yield the result.
Definition 13 (Metric regularity on a set) Let (G 1 , d 1 ) and (G 2 , d 2 ) be metric spaces, and let : The mapping is called metrically regular with gauge When the set V consists of a single point, V = {y}, then is said to be metrically subregular for y on U with gauge μ relative to ⊂ G 1 . When μ is a linear function (that is, μ(t) = κt, ∀t ∈ [0, ∞)), this special case is distinguished as linear metric (sub)regularity with constant κ. When = G 1 , the quantifier "relative to" is dropped. When μ is linear, the infimum of all constants κ for which (17) holds is called the modulus of metric regularity.
The next statement is obvious from the definition. Let (G 1 , d 1 ) and (G 2 , d 2 ) be metric spaces, and let :

Proposition 14
If is metrically subregular with gauge μ at y on U relative to ⊂ G 1 , then is metrically subregular with the same gauge μ at y on all subsets U ⊂ U relative to ⊂ G 1 .

Quantitative convergence
To obtain convergence of fixed point iterations under the assumption of metric subregularity, the gauge of metric subregularity μ is constructed implicitly from another nonnegative function θ : For a p-uniformly convex space the operative gauge satisfies for τ > 0 fixed and θ satisfying (18).
In the case of linear metric subregularity on a CAT(κ) space this becomes If (17) is satisfied for some μ > 0, then the condition μ ≥ τ (1+ ) is satisfied for all μ ≥ μ large enough. The conditions in (18) in this case simplify to θ (t) = γ t, where The next definition characterizes the quantitative convergence of sequences in terms of gauge functions.
(i) (x k ) k∈N is said to be gauge monotone with respect to D with rate μ whenever (ii) (x k ) k∈N is said to be linearly monotone with respect to D with rate c if (21) is satisfied for μ(t) = c · t for all t ∈ R + and some constant c ∈ [0, 1]. A sequence (x k ) k∈N is said to converge gauge monotonically to some element x * ∈ G with rate s k (t) := ∞ j=k μ (j) (t) whenever it is gauge monotone with gauge μ satisfying ∞ j=1 μ (j) (t) < ∞ ∀t ≥ 0, and there exists a constant a > 0 such that d(x k , x * ) ≤ as k (t) for all k ∈ N.
All Fejér monotone sequences are linearly monotone (with constant c = 1) but the converse does not hold (see Proposition 1 and Example 1 of [10]). Gauge-monotonic convergence for a linear gauge in the definition above is just R-linear convergence. Metric subregularity and pointwise (almost) nonexpansiveness are fundamentally connected through the surrogate mapping T S : G → R + ∪ {+∞} defined by where ψ with gauge μ given implicitly by (19) with θ satisfying (18) for τ = (1α)/α and ≥ 0 an upper bound on the violation of pointwise α firmness of T on D. Then, for Moreover, the sequence (x k ) k∈N converges gauge monotonically to some Before proving this theorem, we collect some intermediate results. Proof The assumption that T is pointwise almost α-firmly nonexpansive at all y ∈ Fix T ∩ D with constant α and violation at most on D yields Let x 0 ∈ D and define the sequence x k+1 = Tx k for all k ∈ N. Since T is pointwise almost α-firmly nonexpansive at all points in Fix T ∩ D on D, Fix T ∩ D is closed and P Fix T∩D x k is nonempty (though possibly set-valued) for all k. Denote any selection byx k ∈ P Fix T∩D x k for each k ∈ N. Then On the other hand d( Therefore an iterative application of gauge monotonicity yields Let t 0 = d(x 0 , Fix T ∩ D). For any given natural numbers k, l with k < l, an iterative application of the triangle inequality yields the upper estimate where s k (t 0 ) := ∞ j=k θ (j) (t 0 ) < ∞ for θ satisfying (18). Since (θ (k) (t 0 )) k∈N is a summable sequence of nonnegative numbers, the sequence of partial sums s k (t 0 ) converges to zero monotonically as k → ∞, and hence (x k ) k∈N is a Cauchy sequence and x k → x * for some x * ∈ G. Letting l → +∞ yields In other words, On the other hand, by the assumption that T is pointwise almost α-firmly nonexpansive at all points y ∈ S with the same constant α and violation at most on D we have