Minimal Lipschitz and ∞-harmonic extensions of vector-valued functions on finite graphs

Bacak, Miroslav; Hertrich, Johannes; Neumayer, Sebastian; Steidl, Gabriele

doi:10.1093/imaiai/iaz033

Cited by 2 publications

(2 citation statements)

References 38 publications

(57 reference statements)

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“…In our future research, we will be also interested in L 1 and L ∞ spaces. At least for finite dimensional L ∞ spaces, averaged operators were recently addressed in connection with L ∞ Laplacians in imaging in [18]. The notion of α-firmly nonexpansive operator has also been considered in metric spaces, see for instance [9,10] for the case of a CAT(0) space and [11] for a recent generalization to (non-linear) metric spaces.…”

Section: Introductionmentioning

confidence: 99%

On $$\alpha $$-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

Бeрделлима

Steidl

2021

Results Math

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We introduce the class of $$\alpha $$ α -firmly nonexpansive and quasi $$\alpha $$ α -firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where $$\alpha $$ α -firmly nonexpansive operators coincide with so-called $$\alpha $$ α -averaged operators. For our more general setting, we show that $$\alpha $$ α -averaged operators form a subset of $$\alpha $$ α -firmly nonexpansive operators. We develop some basic calculus rules for (quasi) $$\alpha $$ α -firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) $$\alpha $$ α -firmly nonexpansive. Moreover, we will see that quasi $$\alpha $$ α -firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates $$x_{n+1}:=Tx_{n}$$ x n + 1 : = T x n belong to the fixed point set $${{\,\mathrm{Fix}\,}}T$$ Fix T whenever the operator T is nonexpansive and quasi $$\alpha $$ α -firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in $${{\,\mathrm{Fix}\,}}T$$ Fix T . Further, the projections $$P_{{{\,\mathrm{Fix}\,}}T}x_n$$ P Fix T x n converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in $$L_p$$ L p , $$p \in (1,\infty ) \backslash \{2\}$$ p ∈ ( 1 , ∞ ) \ { 2 } spaces on probability measure spaces.

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Section: Introductionmentioning

confidence: 99%

On $$\alpha $$-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

Бeрделлима

Steidl

2021

Results Math

Self Cite

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show abstract

“…In our future research, we will be also interested in L 1 and L ∞ spaces. At least for finite dimensional L ∞ spaces, averaged operators were recently addressed in connection with L ∞ Laplacians in imaging in [16]. The notion of α-firmly nonexpansive operator has also been considered in metric spaces, see for instance [7], [8] for the case of a CAT(0) space and [9] for a recent generalization to (non-linear) metric spaces.…”

Section: Introductionmentioning

confidence: 99%

On $α$-Firmly Nonexpansive Operators in $r$-Uniformly Convex Spaces

Бeрделлима¹,

Steidl²

2021

Preprint

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We introduce the class of α-firmly nonexpansive and quasi α-firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where α-firmly nonexpansive operators coincide with so-called α-averaged operators. For our more general setting, we show that α-averaged operators form a subset of α-firmly nonexpansive operators. We develop some basic calculus rules for (quasi) α-firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) α-firmly nonexpansive. Moreover, we will see that quasi α-firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder's demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates x n+1 := T x n belong to the fixed point set Fix T whenever the operator T is nonexpansive and quasi α-firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial's property, then these iterates converge weakly to some element in Fix T . Further, the projections P Fix T x n converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in L p , p ∈ (1, ∞)\{2} spaces on probability measure spaces.

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Minimal Lipschitz and ∞-harmonic extensions of vector-valued functions on finite graphs

Cited by 2 publications

References 38 publications

On $$\alpha $$-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

On $$\alpha $$-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

On $α$-Firmly Nonexpansive Operators in $r$-Uniformly Convex Spaces

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