The Cartan–Hadamard Theorem for metric spaces with local geodesic bicombings

Miesch, Benjamin

doi:10.4171/lem/63-1/2-8

Cited by 8 publications

(10 citation statements)

References 10 publications

(17 reference statements)

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“…Indeed, if it were L-Lipschitz for some L ≥ 1, then t = d f (t, t 2 ), f (t, 0) ≤ Ld (t, t 2 ), (t, 0) = Lt 2 , a contradiction. We conclude with an example illustrating some of the properties listed above (see Example 4.2 in [12]).…”

Section: Lipschitz Extensionsmentioning

confidence: 83%

Lipschitz Chain Approximation of Metric Integral Currents

Goldhirsch¹

2021

Preprint

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Section: Lipschitz Extensionsmentioning

confidence: 83%

Lipschitz Chain Approximation of Metric Integral Currents

Goldhirsch¹

2021

Preprint

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“…The first condition on Y , generalizes the existence of a geodesic bicombing, as studied by [96,45,18,113,19], and is analogous in spirit to the idea of a simplicial topological space from homotopy theory [136,Chapter 83.2] and from fuzzy set theory [16], and the peaked partitions of By [68,Theorem 12.1] every subset of R d has the doubling property (see Section 6 for the definition and [68, Section 10.13] for details), and following the discussion on [27, page 3] a metric space is doubling if and only if its metric capacity is finite for all δ ∈ (0, 1].…”

Section: S C : U a Qas Smentioning

confidence: 99%

Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

Acciaio¹,

Kratsios²,

Pammer³

2022

Preprint

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A. We introduce a universal class of geometric deep learning models, called metric hypertransformers (MHTs), capable of approximating any adapted map F : X Z → Y Z with approximable complexity, where X ⊆ R d and Y is any suitable metric space, and X Z (resp. Y Z ) capture all discrete-time paths on X (resp. Y ). Suitable spaces Y include various (adapted) Wasserstein spaces, all Fréchet spaces admitting a Schauder basis, and a variety of Riemannian manifolds arising from information geometry. Even in the static case, where f : X → Y is a Hölder map, our results provide the first (quantitative) universal approximation theorem compatible with any such X and Y . Our universal approximation theorems are quantitative, and they depend on the regularity of F, the choice of activation function, the metric entropy and diameter of X , and on the regularity of the compact set of paths whereon the approximation is performed. Our guiding examples originate from mathematical finance. Notably, the MHT models introduced here are able to approximate a broad range of stochastic processes' kernels, including solutions to SDEs, many processes with arbitrarily long memory, and functions mapping sequential data to sequences of forward rate curves.

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“…Before we start with the proof of Theorem 1.5, we recall some notions from [Mie16]. Let (X, d) be a metric space, let p ∈ X be a point and let r > 0 be a real number.…”

Section: Example 44 We Define the Setmentioning

confidence: 99%

“…In [Mie16], the second named author generalized the classical Cartan-Hadamard Theorem to metric spaces that locally admit a consistent convex geodesic bicombing. With Theorem 1.4 at hand, it is possible to use this generalized Cartan-Hadamard Theorem to obtain the following uniqueness result.…”

Section: Introductionmentioning

confidence: 99%

Conical geodesic bicombings on subsets of normed vector spaces

Basso

Miesch

2019

Advances in Geometry

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In this paper we establish existence and uniqueness results for conical geodesic bicombings on subsets of normed vector spaces. Concerning existence, we give a first example of a convex geodesic bicombing that is not consistent. Furthermore, we show that under a mild geometric assumption on the norm a conical geodesic bicombing on an open subset of a normed vector space locally consists of linear geodesics. As an application, we obtain by the use of a Cartan-Hadamard type result that if a closed convex subset C of a Banach space has non-empty interior, then it admits a unique consistent conical geodesic bicombing, namely the one given by the linear segments.Recently, classical results from the theory of CAT(0) spaces have been transferred to metric spaces that admit conical geodesic bicombings, cf. [Bas17, Des16, DL16, Mie16] and [Kel16]. In the past century, notions related to conical geodesic bicombings have also been considered in metric fixed point theory, most notable W-convexity mappings, cf. [Tak70], and hyperbolic spaces in the sense of S. Reich and I. Shafrir, cf. [RS90]. It is worth to point out that the study of metric spaces that admit conical geodesic bicombing may also lead to new results about word hyperbolic groups, as every word hyperbolic group acts geometrically on a proper, finite dimensional metric space with a unique consistent conical geodesic bicombing (the definitions are given below), cf.[DL15]. The main results of this article show that the several definitions from [DL15] lead to different classes.Our first result deals with convex geodesic bicombings. From now on, we abbreviate D(X) := X × X × [0, 1]. A geodesic bicombing σ : D(X) → X is convex if the map t → d(σ pq (t), σ p q (t)) is convex on [0, 1] for all points p, q, p , q in X. Note that if the underlying metric space is not uniquely geodesic, then a conical geodesic bicombing is not necessarily convex. Examples of conical geodesic bicombings that are not convex are ubiquitous; for instance, non-convex conical geodesic bicombings may be obtained via 1-Lipschitz retractions of linear geodesics, see [DL15, Example 2.2] or Lemma 3.1. In [DL15], it is shown that metric spaces of finite combinatorial dimension in the sense of Dress, cf. [Dre84], possess at most one convex geodesic bicombing. If it exists, this unique convex geodesic bicombing, say σ : D(X) → X, has the property that it is consistent, that is, we have for all points p, q in X that im(σ p q ) ⊂ im(σ pq ) whenever p = σ pq (s) and q = σ pq (t) with 0 ≤ s ≤ t ≤ 1. Clearly, every consistent conical geodesic bicombing is convex. In Section 2, we show that the converse does not hold by proving the subsequent theorem.Theorem 1.1. There is a compact metric space that admits a convex geodesic bicombing which is not consistent.Although there is a non-consistent convex geodesic bicombing on the space considered in Section 2, this space also admits a consistent convex geodesic bicombing. We suspect that this is a general phenomenon. Question 1.2. Let (X, d) be a proper metric spac...

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The Cartan–Hadamard Theorem for metric spaces with local geodesic bicombings

Cited by 8 publications

References 10 publications

Lipschitz Chain Approximation of Metric Integral Currents

Lipschitz Chain Approximation of Metric Integral Currents

Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

Conical geodesic bicombings on subsets of normed vector spaces

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