Geodesic Space Research Articles

Characterization of Gromov-type geodesics

It is well known that, when endowed with the Gromov-Hausdorff distance dGH, the collection M of all isometry classes of compact metric spaces is a complete and separable space. It is also known that (M,dGH) is a geodesic metric space, but there is no known structural characterization of geodesics in M.In this paper we provide two characterizations of geodesics in M. We call a Gromov-Hausdorff geodesic γ:[0,1]→MHausdorff-realizable if there exists a compact metric space Z containing isometric copies of γ(t) for each t∈[0,1] such that the Hausdorff distance satisfies dHZ(γ(s),γ(t))=dGH(γ(s),γ(t)) for all s,t∈[0,1]. In this way, γ is actually a geodesic in the Hausdorff hyperspace of Z, and we call it a Hausdorff geodesic. We prove that in fact every Gromov-Hausdorff geodesic is Hausdorff-realizable. Inspired by this characterization, we further elucidate a structural connection between Hausdorff geodesics and Wasserstein geodesics: we show that every Hausdorff geodesic is equivalent to a so-called Hausdorff displacement interpolation. This equivalence allows us to establish that every Gromov-Hausdorff geodesic is dynamic, a notion which we develop in analogy with dynamic optimal couplings in the theory of optimal transport.Besides geodesics in M, we also study geodesics on the collection Mw of isomorphism classes of compact metric measure spaces. Sturm constructed a family of Gromov-type distances on Mw, which we denote dGW,pS (for p∈[1,∞)), as an analogue of dGH, and proved that (Mw,dGW,pS) is also a geodesic space. We define a notion of Wasserstein-realizabledGW,pS geodesics in a sense similar to Hausdorff-realizable geodesics and show that the set of all Wasserstein-realizabledGW,pS geodesics is dense in the set of all dGW,pS geodesics. We further identify a rich class of dGW,pS geodesics which are Wasserstein-realizable.

Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces

Abstract In this article, we discuss equilibrium problems and their resolvents on complete geodesic spaces. In particular, we consider asymptotic behavior and continuity of resolvents with positive parameter in a complete geodesic space whose curvature is bounded above. Furthermore, we apply these results to resolvents of convex functions.

κ-Galilean and κ-Carrollian noncommutative spaces of worldlines

The noncommutative spacetimes associated to the κ-Poincaré relativistic symmetries and their “non-relativistic” (Galilei) and “ultra-relativistic” (Carroll) limits are indistinguishable, since their coordinates satisfy the same algebra. In this work, we show that the three quantum kinematical models can be differentiated when looking at the associated spaces of time-like worldlines. Specifically, we construct the noncommutative spaces of time-like geodesics with κ-Galilei and κ-Carroll symmetries as contractions of the corresponding κ-Poincaré space and we show that these three spaces are defined by different algebras. In particular, the κ-Galilei space of worldlines resembles the so-called Euclidean Snyder model, while the κ-Carroll space turns out to be commutative. Furthermore, we identify the map between quantum spaces of geodesics and the corresponding noncommutative spacetimes, which requires to extend the space of geodesics by adding the noncommutative time coordinate.

Cross-view learning with scatters and manifold exploitation in geodesic space

<abstract><p>Cross-view data correlation analysis is a typical learning paradigm in machine learning and pattern recognition. To associate data from different views, many approaches to correlation learning have been proposed, among which canonical correlation analysis (CCA) is a representative. When data is associated with label information, CCA can be extended to a supervised version by embedding the supervision information. Although most variants of CCA have achieved good performance, nearly all of their objective functions are nonconvex, implying that their optimal solutions are difficult to obtain. More seriously, the discriminative scatters and manifold structures are not exploited simultaneously. To overcome these shortcomings, in this paper we construct a Discriminative Correlation Learning with Manifold Preservation, DCLMP for short, in which, in addition to the within-view supervision information, discriminative knowledge as well as spatial structural information are exploited to benefit subsequent decision making. To pursue a closed-form solution, we remodel the objective of DCLMP from the Euclidean space to a geodesic space and obtain a convex formulation of DCLMP (C-DCLMP). Finally, we have comprehensively evaluated the proposed methods and demonstrated their superiority on both toy and real datasets.</p></abstract>

Resolvents of the equilibrium problem with generalized perturbations on complete geodesic spaces

Resolvents of the equilibrium problem with generalized perturbations on complete geodesic spaces

Good geodesics satisfying the timelike curvature-dimension condition

Let (M,d,m,≪,≤,τ) be a causally closed, K-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condition wTCDpe(K,N) in the sense of Cavalletti and Mondino. We prove the existence of geodesics of probability measures on M which satisfy the entropic semiconvexity inequality defining wTCDpe(K,N) and whose densities with respect to m are additionally uniformly L∞ in time. This holds apart from any nonbranching assumption. We also discuss similar results under the timelike measure-contraction property.

The Leray-Schauder Principle in Geodesic Spaces

The essential maps introduced by Granas in 1976 is a best tool for proving continuation results for compact maps. Several authors modified this idea to different scenarios. This is a review article, here, we consider the continuation results based on essential maps and the Leray-Schauder principle in the setting of uniquely geodesic spaces [20].

A Gutzwiller Trace Formula for Dirac Operators on a Stationary Spacetime

A Duistermaat–Guillemin–Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch (Adv Math 376:107434, 2021) to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the space of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period.

Gromov Boundaries of Non-proper Hyperbolic Geodesic Spaces

In a proper hyperbolic geodesic space, it is well known that the sequential boundary can be identified as topological spaces with the geodesic boundary. We show that in a (not necessarily proper) hyperbolic geodesic space, the sequential boundary can be identified as topological spaces with the quasi-geodesic boundary.

All noncommutative spaces of κ-Poincaré geodesics

Noncommutative spaces of geodesics provide an alternative way of introducing noncommutative relativistic kinematics endowed with quantum group symmetry. In this paper we present explicitly the seven noncommutative spaces of time-, space- and light-like geodesics that can be constructed from the time-, space- and light-versions of the κ-Poincaré quantum symmetry in (3 + 1) dimensions. Remarkably enough, only for the light-like (or null-plane) κ-Poincaré deformation the three types of noncommutative spaces of geodesics can be constructed, while for the time-like and space-like deformations both the quantum time-like and space-like geodesics can be defined, but not the light-like one. This obstruction comes from the constraint imposed by the coisotropy condition for the corresponding deformation with respect to the isotropy subalgebra associated to the given space of geodesics, since all these quantum spaces are constructed as quantizations of the corresponding classical coisotropic Poisson homogeneous spaces. The known quantum space of geodesics on the light cone is given by a five-dimensional homogeneous quadratic algebra, and the six nocommutative spaces of time-like and space-like geodesics are explicitly obtained as six-dimensional nonlinear algebras. Five out of these six spaces are here presented for the first time, and Darboux generators for all of them are found, thus showing that the quantum deformation parameter κ −1 plays exactly the same algebraic role on quantum geodesics as the Planck constant ℏ plays in the usual phase space description of quantum mechanics.

Principal eigenvalue problem for infinity Laplacian in metric spaces

Abstract This article is concerned with the Dirichlet eigenvalue problem associated with the ∞ \infty -Laplacian in metric spaces. We establish a direct partial differential equation approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the ∞ \infty -eigenvalue problem and show the existence of solutions by adapting Perron’s method. Our method is different from the standard limit process via the variational eigenvalue formulation for p p -Laplacian in the Euclidean space.

Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces

Abstract In this article, we present some new fixed point existence results for nonexpansive-type mappings in geodesic spaces. We also give a number of illustrative examples to settle our claims. We study the asymptotic behavior of Picard iterates generated by these class of mappings under different conditions. Finally, we approximate the solutions of the constrained minimization problem in the setting of Cartan, Alexandrov, and Toponogov (CAT(0)) spaces.

New type parallelogram laws in Banach spaces and geodesic spaces with curvature bounded above

The parallelogram law characterizes a part of the structure of Hilbert spaces. To hold the parallelogram law by the norm, reasonably good conditions are required for Banach spaces. This paper proposes new type parallelogram laws with bifunctions on Banach spaces and geodesic spaces, respectively.

Entropic curvature on graphs along Schrödinger bridges at zero temperature

Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. L{\'e}onard by replacing W2-Wasserstein geodesics by Schr{\"o}odinger bridges in the definition of entropic curvature [23, 25, 24]. As a remarkable fact, as a temperature parameter goes to zero, these Schr{\"o}dinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graph spaces: the lattice Z n endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli-Laplace model. Our general results also apply to a large class of graphs which are not specifically studied in this paper. As opposed to Erbar-Maas results on graphs [27, 10, 11], entropic curvature results of this paper imply new Pr{\'e}kopa-Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube {0, 1} n and for the Bernoulli Laplace model, a new W2 -- W1 transport-entropy inequality is reached, that can not be derived by usual induction arguments over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see [28, 15]) associated to the so-called convex-hull method by Talagrand [38].

Learning isometry-invariant representations for point cloud analysis

3D shape analysis has drawn broad attention due to its increasing demands in various fields. Despite that impressive performance has been achieved on several databases, most researchers focus their efforts on improving the performance of shape classification, retrieval, segmentation, etc. They neglect the fact that the disturbances, such as orientation and deformation, may impact much on the perception, restricting the capacity of generalizing to real applications where the prior of orientation and pose is often unknown. In this paper, we conduct shape analysis on point clouds and propose the point projection feature, which is rotation-invariant. Specifically, a novel architecture is designed to mine features of different levels. We adopt a PointNet-based backbone to extract global feature for the point cloud, and the graph aggregation operation to perceive local pose variance in the Euclidean space or geodesic space. An efficient key point descriptor is designed to assign each point with different response and help recognize the overall geometry. Furthermore, a novel dataset, PKUnon-rigid, is built that is composed of non-rigid 3D objects, based on which we evaluate the capacity of several mainstream methods in terms of processing non-rigid shapes. Mathematical analyses and experimental results demonstrate that the proposed method can extract isometry-invariant representations for 3D shape analysis tasks without rotation augmentation, and outperforms other state-of-the-art methods. The proposed dataset is publicly available at https://github.com/tasx0823/PKUnon-rigid.

Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

Abstract Given a polarized projective variety ( X , L ) {(X,L)} over any non-Archimedean field, assuming continuity of envelopes, we define a metric on the space of finite-energy metrics on L, related to a construction of Darvas in the complex setting. We show that this makes finite-energy metrics on L into a geodesic metric space, where geodesics are given as maximal psh segments. Given two continuous psh metrics, we show that the maximal segment joining them is furthermore continuous. Our results hold in particular in all situations relevant to the study of degenerations and K-stability in complex geometry.

Approximating Solutions of Optimization Problems via Fixed Point Techniques in Geodesic Spaces

The principal objective of this paper is to find the solution to a constrained minimization problem and the zero of the monotone operator in geodesic spaces. The basic tool in our study is a nonexpansive mapping. Further, we employ the general Picard–Mann iterative method to approximate fixed points of nonexpansive mappings under various conditions. We obtain certain theorems concerning Δ and strong convergence.

The Hajłasz Capacity Density Condition is Self-improving

We prove a self-improvement property of a capacity density condition for a nonlocal Hajłasz gradient in complete geodesic spaces with a doubling measure. The proof relates the capacity density condition with boundary Poincaré inequalities, adapts Keith–Zhong techniques for establishing local Hardy inequalities and applies Koskela–Zhong arguments for proving self-improvement properties of local Hardy inequalities. This leads to a characterization of the Hajłasz capacity density condition in terms of a strict upper bound on the upper Assouad codimension of the underlying set, which shows the self-improvement property of the Hajłasz capacity density condition.

Contractions in Persistence and Metric Graphs

We prove that the existence of a $1$-Lipschitz retraction (a contraction) from a space $X$ onto its subspace $A$ implies the persistence diagram of $A$ embeds into the persistence diagram of $X$. As a tool we introduce tight injections of persistence modules as maps inducing the said embeddings. We show contractions always exist onto shortest loops in metric graphs and conjecture on existence of contractions in planar metric graphs onto all loops of a shortest homology basis. Of primary interest are contractions onto loops in geodesic spaces. These act as ideal circular coordinates. Furthermore, as the Theorem of Adamaszek and Adams describes the pattern of persistence diagram of $S^1$, a contraction $X \to S^1$ implies the same pattern appears in persistence diagram of $X$.

On the logarithmic coarse structures of Lie groups and hyperbolic spaces

We characterize the Lie groups with finitely many connected components that are O(u)-bilipschitz equivalent (almost quasiisometric in the sense that the sublinear function u replaces the additive bounds of quasiisometry) to the real hyperbolic space, or to the complex hyperbolic plane. The characterizations are expressed in terms of deformations of Lie algebras and in terms of pinching of sectional curvature of left-invariant Riemannian metrics in the real case. We also compare sublinear bilipschitz equivalence and coarse equivalence, and prove that every coarse equivalence between the logarithmic coarse structures of geodesic spaces is a \(O(\log )\)-bilipschitz equivalence. The Lie groups characterized are exactly those whose logarithmic coarse structure is equivalent to that of a real hyperbolic space or the complex hyperbolic plane. Finally we point out that a conjecture made by Tyson about the conformal dimensions of the boundaries of certain hyperbolic buildings holds conditionally to the four exponentials conjecture.