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 “Geodesic Optimization”
Minimization of Functions on Dually Flat Spaces Using Geodesic Descent Based on Dual Connectionsstat.CO
Geodesic descent optimizes likelihood in dually flat spaces.
problem Maximum likelihood estimation in exponential families.
method m-geodesic and e-geodesic updates on dually flat spaces.
result Geodesic updates can reach maximum likelihood estimator in one step.
Optimizing quantum graphs yields geodesic nets on surfaces.
problem Finding optimal quantum graphs for geodesic nets.
method Optimizing functionals from spectral theory to find geodesic nets.
result Critical metrics for eigenvalues give rise to geodesic nets.
Optimal geodesics connect boundary points in Teichmüller space.
problem Finding optimal geodesics between boundary points of Teichmüller space.
method Analyzing horofunctions and Teichmüller geodesics.
result There is a unique optimal geodesic connecting boundary points.
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 paper constructs optimal sub-Riemannian geodesics in specific Carnot groups.
problem Optimal paths in sub-Riemannian geometry for certain groups.
method Explicit construction of geodesics using symmetries and the Hadamard technique.
result Identification of cut time and cut locus in the constructed geodesics.
Generates samples conditioned on labels using optimal transport.
problem Estimating conditional distributions for specific labels.
method Wasserstein geodesic generator based on optimal transport theory.
result Learned conditional distributions and optimal transport maps.
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.
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…
Neural solver computes Wasserstein geodesics and velocity fields efficiently.
problem Computing Wasserstein geodesics and velocity fields efficiently.
method Sample-based neural network approach to solve the minimax problem.
result Directly samples from target distribution and estimates velocity field.
Study of Gödel Universe as Lie group with specific metric.
problem Characterize geodesics in the Gödel Universe.
method Geometric theory of optimal control applied to Lie groups with left-invariant Lorentz metrics.
result No closed timelike or isotropic geodesics in the Gödel Universe.
Geodesic completeness and optimal Sobolev index proven for Minkowski spacetimes.
problem Geodesic completeness and optimal Sobolev index for Minkowski spacetimes.
method Null non-trapping condition and real principal type estimate.
result Optimal Sobolev index proven for asymptotically Minkowski spacetimes.
ResNets learn the geodesic curve in Wasserstein space.
problem Characterize the dynamics of deep residual networks during training.
method Modeling ResNet dynamics using continuity equations and optimal transport.
result ResNets learn the geodesic curve in the Wasserstein space.
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.
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.
The paper analyzes symmetries of Vaidya-Bonner geodesics.
problem Investigating invariance properties of Vaidya-Bonner geodesics.
method Classification of Lie point symmetries and Noether symmetries, determination of optimal system of subalgebras.
result Determination of optimal system of subalgebras for Vaidya-Bonner geodesics.
Optimizes Euclidean functions on Riemannian manifolds with warped metrics.
problem Optimizing functions in high-dimensional Euclidean spaces.
method Riemannian geometry, warped metric, geodesic curves, Taylor approximations, retraction maps.
result Efficient optimization of functions using third-order approximations of geodesics.
New quasi-geodesics for Stiefel manifold simplify complex computations.
problem Efficiently solving geodesic endpoint problem on Stiefel manifold.
method Derived new representations of quasi-geodesics for large-scale computations.
result New quasi-geodesics are closer to Riemannian geodesics.
Lower bounds on geodesic lengths for spheres with Willmore energy.
problem Finding shortest closed geodesics on spheres with Willmore energy.
method Proving a lower bound on geodesic lengths for spheres with Willmore energy below 6π.
result The energy threshold of 6π is optimal and the inequality cannot be extended to higher genus surfaces.
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 …
Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spacesmath.MG
We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal ma…
Optimizes tensor completion using geodesics on Segre manifolds.
problem Incomplete tensor data in recommender systems and spectroscopy.
method Riemannian conjugate gradient optimization with explicit geodesic expressions.
result Recovery of tensor decomposition from as little as 10% of data.
New optimality conditions for sub-Riemannian geodesics derived.
problem Optimality conditions for sub-Riemannian geodesics.
method Geometric translation and ODE derivation.
result New second-order necessary optimality conditions.
Sharp bounds found on shortest geodesic on punctured spheres.
problem Finding the shortest closed geodesic on punctured spheres.
method Sharp curvature-free upper bounds expressed in terms of area, extremal metrics described.
result Optimal bounds for spheres with up to four ends, extended to larger numbers of punctures.
We show that every closed Lorentzian surface contains at least two closed geodesics. Explicit examples show the optimality of this claim. Refining this result we relate the least number of closed geodesics to the causal structure of the surface and the homotopy type of the Lorentzian metric.
Hedlund constructed Riemannian metrics on n-tori, for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics are optimal for nilpotent fundamental groups.
The paper proves a conjecture about the minimum number of closed geodesics on a Finsler 3-sphere.
problem Proving the Anosov conjecture for bumpy Finsler 3-spheres.
method Analyzing the Morse index of prime closed geodesics.
result Established the conjecture for Finsler 3-spheres with nonzero Morse index.
We create a smooth manifold of triangular meshes with a geodesically complete metric.
problem Representing and manipulating 2D shapes as triangular meshes.
method Developed a geodesically complete Riemannian metric for triangular meshes.
result The metric preserves mesh connectivity and avoids mesh degradation.
A simple model for unbalanced optimal transport captures key features.
problem Capturing the main features of unbalanced optimal transport.
method Introducing a metric on the conical extension of diffeomorphisms and studying its properties.
result Total mass evolves with constant acceleration along geodesics.
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.
On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on L…
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.
GeONet learns the Wasserstein geodesic without mesh discretization.
problem Computing the Wasserstein geodesic between complex data distributions.
method Mesh-invariant deep neural operator network that learns saddle point optimality conditions.
result GeONet achieves comparable accuracy to standard OT solvers with reduced computational cost.
GEORCE computes geodesics quickly and accurately.
problem Computing geodesics on Riemannian and Finsler manifolds is difficult and inefficient.
method GEORCE transforms geodesic computation into a discrete control problem.
result GEORCE achieves global convergence and quadratic local convergence.
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.
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…
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.
Study on geodesics in Cartan group sub-Riemannian problem, proving conjugate time relation to Maxwell time.
problem Geodesics in sub-Riemannian problem on Cartan group.
method Analysis of symmetries, geodesic optimality, conjugate time calculation.
result First conjugate time is not less than Maxwell time, and equal for certain geodesics.
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.
NR retraction approximates geodesics on submanifolds efficiently.
problem Efficiently approximating geodesics on submanifolds for practical algorithms.
method Introducing Newton retraction (NR) as a class of retractions on submanifolds induced by a foliation of the ambient manifold.
result NR is more stable and computationally cheaper than oblique projection, with superlinear convergence regions.
A new method for fast optimal transport using sliced Wasserstein generalized geodesics.
problem Computing optimal transport distances efficiently and accurately.
method Proposes a new proxy of squared Wasserstein distance based on one-dimensional projections.
result min-SWGG is an upper bound of Wasserstein distance with similar computational complexity.
Introduces PCG for better counterfactual explanations in vision models.
problem Ambiguity in latent-space optimization methods for counterfactual explanations.
method Constructs counterfactuals by tracing geodesics under a perceptually Riemannian metric.
result PCG outperforms baselines and reveals hidden failure modes.
We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalisations of the --H…
Geodesics on Grushin spacesmath.DG
Study geodesics on Grushin spaces, proving upper bounds on conjugate times.
problem Classify geodesics on higher-dimensional Grushin spaces.
method Solve Hamilton's equations using calculus of generalized trigonometric functions, analyze symmetries, and use density arguments.
result Prove a conjectured cut time provides an upper bound on conjugate times.
New approach finds minima of geodesic lengths for non-uniform fillings.
problem Finding minima of geodesic length functions for non-uniform fillings.
method Elementary optimization for 4-regular topological fillings, analysis of fat graphs and optimization techniques.
result Minima of geodesic length functions are found to be at triangle surfaces in both analyzed classes of non-uniform fillings.
In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on . These structures are connected with each other, and it is possible to use information about one of them to obtain results about another one. We give an explicit parameterization of sub-Riemannian g…
Study local control in a 7D quaternionic Heisenberg group.
problem Optimizing geodesics in a 7D quaternionic Heisenberg group.
method Matrix representation and analysis of sub-Riemannian structure symmetries.
result Impact of symmetries on geodesic optimality.