Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined by how it interacts with the standard differential structure in this space -- the…
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48 results for “geodesically convex optimization”
Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for …
Extends DCP framework to Hadamard manifolds for geodesically convex functions.
problem Verifying convexity in nonlinear programs on Hadamard manifolds.
method Introduces Disciplined Geodesically Convex Programming (DGCP) framework, defining compositions and transformations for geodesically convex functions.
result Allows verification of geodesic convexity for a broader range of functions, including statistical estimators and matrix-valued optimization.
New study shows acceleration in hyperbolic spaces is impossible for strongly geodesically convex functions.
problem Acceleration in hyperbolic spaces for strongly geodesically convex functions is impossible.
method Perturbing hard functions with sums of bump functions chosen by a resisting oracle.
result Acceleration is unachievable for any deterministic algorithm in hyperbolic spaces for strongly geodesically convex functions.
Riemannian algorithms converge at Euclidean rates for geodesically convex-concave problems.
problem Min-max optimization on Riemannian manifolds.
method RCEG method and RGDA for geodesically strongly-convex-concave problems.
result RCEG achieves linear convergence rate in geodesically strongly-convex-concave cases.
Lower bounds for geodesically convex optimization show curvature negatively impacts complexity.
problem Understanding the impact of curvature on the query complexity of geodesically convex optimization.
method Building on recent lower bounds, the study proposes and proves new lower bounds for various settings of geodesically convex optimization.
result Negative curvature is detrimental to the complexity of geodesically convex optimization.
New method tackles geodesically convex optimization with polynomial convergence.
problem Designing an efficient algorithm for geodesically convex optimization.
method Ellipsoid-like algorithm with polynomial query and per-query complexity.
result Achieves polynomial convergence for geodesically convex functions.
Paper tackles online learning on curved spaces without projections.
problem Online learning on Riemannian manifolds with computational constraints.
method Develops projection-free algorithms for geodesically convex optimization.
result Achieves sub-linear regret guarantees in online geodesically convex optimization.
We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.
New method for optimization on Hadamard manifolds with curvature-independent guarantees.
problem Curvature-dependent complexity in geodesic convex optimization.
method Introducing horospherical convexity and developing algorithms for optimization.
result Curvature-independent convergence of subgradient descent and Nesterov's method.
Paper proposes a new classifier for hyperbolic spaces using horospherical boundaries.
problem Optimization of large margin classifiers in hyperbolic spaces.
method Horospherical decision boundaries for geodesically convex optimization.
result Geodesically convex optimization leads to globally optimal solutions.
As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic -convex function and deduce some basic properties of -convex function and geodesic -convex function. We also introduce the concept of geodesic -convex set and -epigraph and in…
New methods optimize functions on hyperbolic and spherical spaces, matching Euclidean rates up to logarithmic factors.
problem Optimizing functions on non-Euclidean spaces like hyperbolic and spherical geometries.
method Introduced accelerated global first-order methods for -smooth and geodesically convex functions on hyperbolic and spherical spaces.
result Achieved the same rates as accelerated gradient descent in Euclidean space, up to logarithmic factors.
Self-crossing geodesicsmath.DG
Self-crossing geodesics on convex surfaces are studied.
problem Understanding patterns of geodesics crossing themselves.
method Analyzing closed geodesics on convex surfaces.
result Self-crossing geodesics exist on convex surfaces.
IPMs struggle with hyperbolic spaces due to polynomially growing barrier parameters.
problem IPMs' efficiency is hindered in hyperbolic spaces.
method Analyzing the barrier parameter growth in hyperbolic and Hadamard spaces.
result The barrier parameter grows polynomially with the domain's diameter in hyperbolic spaces.
New proof of energy functional monotonicity via geodesics in measure space.
problem Proving monotonicity of energy functional in generalized Ricci flow.
method Defining adapted cost functional, geodesics, and entropy functional.
result Monotonicity of cost along backwards heat flow and energy functional along generalized Ricci flow.
The authors define a class of functions on Riemannian manifolds, which is called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming i…
Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometrymath.OC
The paper solves optimal control problems for various convex sets using convex trigonometry.
problem Optimal control problems with 2D convex compact sets.
method Using convex trigonometry to derive extremals for various problems.
result Geodesics in multiple sub-Finsler problems are derived.
New global section found for geodesic flows on convex hypersurfaces.
problem Finding global sections for geodesic flows on convex hypersurfaces.
method Constructing a global hypersurface of section with an isometric involution.
result Generalized Birkhoff annulus to higher dimensions.
Geodesic flow mixing on convex projective manifolds proven.
problem Understanding mixing properties of geodesic flow on convex projective manifolds.
method Introduced biproximal unit tangent bundle and proved mixing properties.
result Geodesic flow is topologically mixing on biproximal unit tangent bundle.
Stable nets on convex hypersurfaces maintain their shape under small perturbations.
problem Maintaining the shape of nets on convex surfaces under slight changes.
method Constructing stable geodesic nets on convex hypersurfaces.
result Stable geodesic nets on convex hypersurfaces do not change shape under small perturbations.
Study on Mabuchi functional's convexity using ε-geodesics.
problem Understanding the convexity of the Mabuchi functional.
method Analysis of ε-geodesics to study the Mabuchi functional's convexity.
result Uniform fiberwise non-degeneracy of geodesics when Mabuchi functional is ε-affine.
No accelerated gradient method for hyperbolic convex functions.
problem Existence of accelerated gradient methods for geodesically convex functions on hyperbolic spaces.
method Analysis of volume growth in negatively curved spaces.
result No-go theorem for accelerated gradient methods on hyperbolic plane.
Stability of Convex Spheresmath.DG
Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
A new method solves convex optimization problems on manifolds efficiently.
problem Optimization on Hadamard manifolds with convex objectives.
method Intrinsic Riemannian proximal gradient method.
result Sublinear and linear convergence rates for convex and strongly convex problems, respectively.
Develops iso-Riemannian optimization for data manifolds.
problem Challenges in performing optimization on learned data manifolds.
method Introduces iso-connection and iso-Riemannian descent algorithm.
result Demonstrates efficient solutions to inverse problems on learned data manifolds.
Entropy study of geodesic flow on convex projective surfaces.
problem Entropy of Sinai-Ruelle-Bowen measure on convex projective surfaces.
method Analysis of Hilbert area and Blaschke metric.
result Entropy tends to zero if and only if the Hilbert area tends to infinity.
This paper explores the relation between convex functions and the geometry of space-times and semi-Riemannian manifolds (an investigation initiated by Gibbons-Ishibashi). Specifically, we study geodesic connectedness. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which kno…
New method optimizes on curved manifolds without curvature dependence.
problem Curvature-dependent regret in online optimization on Hadamard manifolds.
method Riemannian online gradient descent for h-convex functions.
result Established and regret guarantees, curvature-independent.
The study finds at least two short, simple geodesic chords on a disk with convex boundary.
problem Existence of short, simple geodesic chords on a 2-disk with convex boundary.
method Proof of existence using Riemannian geometry and bounds on lengths.
result Existence of at least two short, simple orthogonal geodesic chords on a 2-disk with convex boundary.
Asymptotic geodesics in convex polygons are convex for large distances.
problem Understanding convexity of geodesics in Hilbert geometry.
method Analyzing the distance function between asymptotic geodesics for large t.
result The distance function between asymptotic geodesics is convex for sufficiently large t.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…
Geodesic flows on specific manifolds are structurally stable.
problem Stability of geodesic flows on compact manifolds without conjugate points.
method Analyzing the compact manifold with quasi-convex universal covering and divergent geodesic rays.
result Proved the -stability conjecture for geodesic flows of compact manifolds.
Geodesically convex functions are continuous on Riemannian manifolds.
problem Continuity of geodesically convex functions on Riemannian manifolds.
method Proof of continuity using geodesic convexity and addressing a gap in existing proof.
result All geodesically convex functions are continuous in the interior of their domain on Riemannian manifolds.
The paper proves the existence and properties of geodesics on convex surfaces.
problem Existence and properties of geodesics on convex surfaces with free boundaries.
method Free boundary curve shortening flow on closed surfaces with strictly convex boundary.
result Existence of two free boundary embedded geodesics and geodesics with Morse Index 1 and 2.
We prove that in metric measure spaces where the entropy functional is K-convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one opti…
We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…
Develops a Riemannian archetypal analysis for interpretable non-linear data.
problem Limited performance of classical archetypal analysis on non-linear data.
method Riemannian geometry for data-driven pullback, geodesic convex combinations, convex relaxation followed by non-convex refinement.
result Combines interpretability of classical archetypal analysis with expressive power of modern non-linear models.
Geodesics in Kähler metrics connect metrics with constant scalar curvature.
problem Deriving geodesics for relatively Kähler metrics on fibrations.
method Deriving geodesic equation, proving uniqueness, convexity of log-norm functional.
result Fibrations with optimal symplectic connections are polystable.
In non-compact manifolds, geodesic flowers exist.
problem Existence of geodesic flowers in non-compact manifolds.
method Proving the existence of non-trivial geodesic flowers in complete non-compact manifolds with locally convex ends.
result Non-trivial geodesic flowers exist in every complete non-compact manifold with locally convex ends.
New constructions show stable geodesics and figure-eights in convex hypersurfaces.
problem Constructing stable geodesics and figure-eights in convex hypersurfaces.
method Explicit billiard trajectories with controlled parallel transport in convex polytopes.
result Construction of stable figure-eights and index-zero geodesics in convex hypersurfaces.
Billiard trajectories and geodesics are closely related geometrically.
problem Understanding the relationship between billiard trajectories and geodesics on surfaces.
method Establishing mutual approximation results for billiard trajectories and geodesic segments on surfaces.
result For Riemannian billiard tables, there are families of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory.
In this paper we have generalized the notion of -radial contraction in complete Riemannian manifold and developed the concept of -convex function. We have also given a counter example proving the fact that in general -radial contraction of a geodesic is not necessarily a geodesic. We have also deduced some r…
We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron.
Strong geodesic convex function and strong monotone vector field of order on Riemannian manifolds have been established. A characterization of strong geodesic convex function of order for the continuously differentiable functions has been discussed. The relation between the solution of a new variational inequal…
Study geodesic distances and convexity in contact sets.
problem Understanding geodesic distances and convexity in contact sets.
method Extending results on quasi-psh functions and big cohomology classes, studying Monge-Ampère measures on contact sets.
result Convexity of the K-energy in big and nef cohomology classes.