A new proof of the Lie–Trotter–Kato formula in Hadamard spaces

Bacak, Miroslav

doi:10.1142/s0219199713500442

Cited by 5 publications

(5 citation statements)

References 38 publications

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“…x, x ∈ dom f, for every t ∈ [0, ∞). Gradient flow semigroups in Hadamard spaces have been studied by several authors [18,21,26,4,5,6] and the theory can be extended to more general metric spaces [1].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces

Bacak

Reich

2014

J. Fixed Point Theory Appl.

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Abstract. We study a nonlinear semigroup associated with a nonexpansive mapping on an Hadamard space and establish its weak convergence to a fixed point. A discretetime counterpart of such a semigroup, the proximal point algorithm, turns out to have the same asymptotic behavior. This complements several results in the literatureboth classical and more recent ones. As an application, we obtain a new approach to heat flows in singular spaces for discrete as well as continuous times.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces

Bacak

Reich

2014

J. Fixed Point Theory Appl.

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“…. , C N ⊂ H. Since the distance functions to such sets are convex continuous (see Example 1.3 below) in Hadamard spaces, the objective function in this problem is of the form (3). Namely, we are to minimize the function (5) f…”

Section: Introductionmentioning

confidence: 99%

“…The proof given in [55] uses ultralimits of Hadamard spaces. A simpler proof relying on weak convergence appeared in [3].…”

Section: Introductionmentioning

confidence: 99%

Computing Medians and Means in Hadamard Spaces

Bacak¹

2014

SIAM J. Optim.

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155

164

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Abstract. The geometric median as well as the Fréchet mean of points in an Hadamard space are important in both theory and applications. Surprisingly, no algorithms for their computation are hitherto known. To address this issue, we use a split version of the proximal point algorithm for minimizing a sum of convex functions and prove that this algorithm produces a sequence converging to a minimizer of the objective function, which extends a recent result of D. Bertsekas (2001) into Hadamard spaces. The method is quite robust and not only does it yield algorithms for the median and the mean, but it also applies to various other optimization problems. We moreover show that another algorithm for computing the Fréchet mean can be derived from the law of large numbers due to K.-T. Sturm (2002).In applications, computing medians and means is probably most needed in tree space, which is an instance of an Hadamard space, invented by Billera, Holmes, and Vogtmann (2001) as a tool for averaging phylogenetic trees. It turns out, however, that it can be also used to model numerous other tree-like structures. Since there now exists a polynomial-time algorithm for computing geodesics in tree space due to M. Owen and S. Provan (2011), we obtain efficient algorithms for computing medians and means, which can be directly used in practice.

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“…Moreover, along the lines of [Ma] and [CM], we study the large time behavior of the flow ( §4.5) and prove a Trotter-Kato product formula for pairs of semi-convex functions (Theorem 5.4). The latter is a two-fold generalization of the existing results in [CM], [Sto] and [Ba1] for convex functions on CAT(0)-spaces (to be precise, an inequality corresponding to our key estimate with λ = 0 and K = 2 is an assumption of [CM]). We stress that we use only the qualitative properties of CAT(1)-spaces instead of the direct curvature condition.…”

Section: Introductionmentioning

confidence: 73%

“…See [KM] for the classical setting of convex functions on Hilbert spaces. The Trotter-Kato product formula on metric spaces was established by Stojkovic [Sto] for convex functions on CAT(0)-spaces in terms of ultralimits (see also a recent result [Ba1] in terms of weak convergence), and by Clément and Maas [CM] for functions satisfying the assertion of our key lemma (Lemma 3.1) with K = 2 and λ = 0 (thus including convex functions on CAT(0)-spaces). We stress that, similarly to the previous section, both the squared distance function and potential functions are allowed to be semi-convex in our argument.…”

Section: A Trotter-kato Product Formulamentioning

confidence: 99%

Gradient flows and a Trotter-Kato formula of semi-convex functions on CAT(1)-spaces

Ohta¹,

Pálfia²

2017

American Journal of Mathematics

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We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio-Gigli-Savaré, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativity with semi-convex squared distance functions. Our approach combining the semiconvexity of the squared distance function with a Riemannian property of the space seems to be of independent interest, and can be compared with Savaré's work on the local angle condition under lower curvature bounds. Applications include the convergence of the discrete variational scheme to a unique gradient curve, the contraction property and the evolution variational inequality of the gradient flow, and a Trotter-Kato product formula for pairs of semi-convex functions.

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A new proof of the Lie–Trotter–Kato formula in Hadamard spaces

Cited by 5 publications

References 38 publications

The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces

The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces

Computing Medians and Means in Hadamard Spaces

Gradient flows and a Trotter-Kato formula of semi-convex functions on CAT(1)-spaces

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