Global Nonpositive Curvature Research Articles

Characterizations of higher rank hyperbolicity

In analogy to the various characterizations of Gromov hyperbolicity, we present a list of six mutually equivalent higher rank conditions for metric spaces satisfying some assumption reminiscent of global non-positive curvature.

Tracking and Regret Bounds for Online Zeroth-Order Euclidean and Riemannian Optimization

We study numerical optimization algorithms that use zeroth-order information to minimize time-varying geodesically convex cost functions on Riemannian manifolds. In the Euclidean setting, zeroth-order algorithms have received a lot of attention in both the time-varying and time-invariant cases. However, the extension to Riemannian manifolds is much less developed. We focus on Hadamard manifolds, which are a special class of Riemannian manifolds with global nonpositive curvature that offer convenient grounds for the generalization of convexity notions. Specifically, we derive bounds on the expected instantaneous tracking error, and we provide algorithm parameter values that minimize the algorithm's performance. Our results illustrate how the manifold geometry in terms of the sectional curvature affects these bounds. Additionally, we provide dynamic regret bounds for this online optimization setting. To the best of our knowledge, these are the first regret bounds even for the Euclidean version of the problem. Lastly, via numerical simulations, we demonstrate the applicability of our algorithm on an online Karcher mean problem.

L-CONVEXITY ON GRAPH STRUCTURES

In this paper, we study classes of discrete convex functions: submodular functions on modular semilattices and L-convex functions on oriented modular graphs. They were introduced by the author in complexity classification of minimum 0-extension problems. We clarify the relationship to other discrete convex functions, such as k-submodular functions, skew-bisubmodular functions, L♮-convex functions, tree submodular functions, and UJ-convex functions. We show that they actually can be viewed as submodular/L-convex functions in our sense. We also prove a sharp iteration bound of the steepest descent algorithm for minimizing our L-convex functions. The underlying structures, modular semilattices and oriented modular graphs, have rich connections to projective and polar spaces, Euclidean building, and metric spaces of global nonpositive curvature (CAT(0) spaces). By utilizing these connections, we formulate an analogue of the Lovász extension, introduce well-behaved subclasses of submodular/L-convex functions, and show that these classes can be characterized by the convexity of the Lovász extension. We demonstrate applications of our theory to combinatorial optimization problems that include multicommodity flow, multiway cut, and related labeling problems: these problems had been outside the scope of discrete convex analysis so far.

Packing and Covering with Balls on Busemann Surfaces

In this note we prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $\delta$, the minimum number of closed balls of radius $\delta$ with centers at $\mathcal S$ and covering the set $S$ is at most 19 times the maximum number of disjoint closed balls of radius $\delta$ centered at points of $S$: $\nu(S) \le \rho(S) \le 19\nu(S)$, where $\rho(S)$ and $\nu(S)$ are the covering and the packing numbers of $S$ by ${\delta}$-balls.

A Good Earthquake Concave Behaviour of a Seismic Isolator Which Supports a Metallic Roof

We combine some important theoretical mathematical results with practical aspects from engineering in a nice framework of K-spiders, which forms a space with global nonpositive curvature. Some concavity properties into such framework are derived. As an application, we consider a mathematical model for an engineering problem. More precisely, we are trying to model the behaviour of tension forces appearing in a metallic roof supported on seismic isolators. In fact, we are looking to find the properties of good seismic isolators in order to reduce the destructive power of an earthquake. The answer consists of a concave behaviour in terms of the displacements on the 3D axes of the Cartesian system.

Stochastic Aspects of Nonlinear Refinement Schemes

This article develops a stochastic viewpoint on nonlinear refinement algorithms designed to construct continuous objects from discrete samples in a global NPC (nonpositive curvature) space. A particular focus lies on convergence properties of so-called barycentric subdivision schemes, a class of refinement algorithms acting on input data via weighted averages. An essential observation is the characterization of these kinds of schemes as nonlinear Markov semigroups. Exploiting this feature, it is proven that convergence on arbitrary NPC spaces is equivalent to convergence for real-valued input data. Furthermore, a strong law of large numbers leads to certain structure-preserving properties for barycentric schemes on the space of positive definite matrices. A concluding section addresses the relationship between the convergence properties of a scheme and its so-called characteristic Markov chain.

Shortest path problem in rectangular complexes of global nonpositive curvature

CAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path γ(x,y). In this paper, we present an efficient algorithm for answering two-point distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons (CAT(0) rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs (CAT(0) rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees ⩾4). Namely, we show that for a CAT(0) rectangular complex K with n vertices, one can construct a data structure D of size O(n2) so that, given any two points x,y∈K, the shortest path γ(x,y) between x and y can be computed in O(d(p,q)) time, where p and q are vertices of two faces of K containing the points x and y, respectively, such that γ(x,y)⊂K(I(p,q)) and d(p,q) is the distance between p and q in the underlying graph of K. If K is a ramified rectilinear polygon, then one can construct a data structure D of optimal size O(n) and answer two-point shortest path queries in O(d(p,q)logΔ) time, where Δ is the maximal degree of a vertex of G(K). Finally, if K is a squaregraph, then one can construct a data structure D of size O(nlogn) and answer two-point shortest path queries in O(d(p,q)) time.

A version of the Hermite--Hadamard inequality in a nonpositve curvature space

We obtain some Hermite{Hadamard type inequalities for convex functions in a global non-positive curvature space.

Geodesics in CAT(0) cubical complexes

We describe an algorithm to compute the geodesics in an arbitrary CAT(0) cubical complex. A key tool is a correspondence between cubical complexes of global non-positive curvature and posets with inconsistent pairs. This correspondence also gives an explicit realization of such a complex as the state complex of a reconfigurable system, and a way to embed any interval in the integer lattice cubing of its dimension.

Fan’s inequality in geodesic spaces

Fan’s minimax inequality is extended to the context of metric spaces with global nonpositive curvature. As a consequence, a much more general result on the existence of a Nash equilibrium is obtained.

The Hermite–Hadamard inequality for convex functions on a global NPC space

We prove an extension of Choquet's theorem to the framework of compact metric spaces with a global nonpositive curvature. Together with Sturm's extension [K.T. Sturm, Probability measures on metric spaces of nonpositive curvature, in: Pascal Auscher, et al. (Eds.), Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs April 16–July 13, 2002, Paris, France, in: Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357–390] of Jensen's inequality, this provides a full analogue of the Hermite–Hadamard inequality for the convex functions defined on such spaces.

Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature

AbstractThe Schauder fixed point theorem is extended to the context of metric spaces with global nonpositive curvature. Some applications are included.