We consider a left invariant Riemannian metric on SO(3) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus …
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48 results for “cut time”
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Understanding Riemannian metrics on lens spaces and their geometric properties.
method Geometric control theory methods applied to axisymmetric metrics.
result Cut loci and cut times converge to sub-Riemannian structure's values.
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Analyzing Riemannian metrics on lens spaces.
method Geometric control theory methods.
result Cut loci and cut times converge to sub-Riemannian structure's cut locus and time.
We consider the Lie group PSL(2) (the group of orientation preserving isometries of the hyperbolic plane) and a left-invariant Riemannian metric on this group with two equal eigenvalues that correspond to space-like eigenvectors (with respect to the Killing form). For such metrics we find a parametrization of geodesics…
In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest 2-dimensional Riemannian manifolds different from the sphere with non trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate poi…
Stability of cut locus under metric perturbations in compact Riemannian manifolds.
problem Stability of cut locus under -perturbations of the metric.
method Proving stability with respect to the Hausdorff metric of the cut locus under perturbation of the metric.
result The Hausdorff distance between cut loci converges to zero as the metrics converge.
The paper proves Lipschitz continuity of cut times in spacetimes.
problem Lipschitz continuity of cut times in globally hyperbolic spacetimes.
method Adapted Itoh-Tanaka method to Lorentzian setting.
result Lipschitz continuity of cut times with quantitative estimates.
Study of longest arcs and cut loci in deformed anti de-Sitter spaces.
problem Existence and properties of time-like cycles in deformed Lorentzian manifolds.
method Analysis of universal covering, admissible curves, and Lorentzian geodesics.
result Identification of cut time and cut locus in deformed anti de-Sitter spaces.
Reduced sub-Riemannian time on a specific group structure.
problem Optimizing paths in a sub-Riemannian structure on a Carnot group.
method Proved conjectured cut times, compared with known results, and solved equations in elliptic functions.
result Reduced cut times for sub-Riemannian paths on the Cartan group.
Financial time series have been investigated to follow fat-tailed distributions. Further, an empirical probability distribution sometimes shows cut-off shapes on its tails. To describe this stylized fact, we incorporate the cut-off effect in superstatistics. Then we confirm that the presented stochastic model is capabl…
Differentiable cutting-plane layers solve parametric mixed-integer linear optimization problems.
problem Solving parametric mixed-integer linear optimization problems with changing data.
method Introducing cutting-plane layers (CPLs) for differentiable cutting-plane generation.
result The algorithm computes solutions with low integrality gaps and generalizes to unseen instances.
In this note, we study the cut locus of the free, step two Carnot groups with generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Myasnichenko - 2002] and [Montanari, Morbidelli - 2016], by exh…
We study the small time asymptotics of the gradient and Hessian of the logarithm of the heat kernel at the cut locus, giving, in principle, complete expansions for both quantities. We relate the leading terms of the expansions to the structure of the cut locus, especially to conjugacy, and we provide a probabilistic in…
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.
The study examines optimal synthesis in a radially symmetric Grushin space with conditions on the weight function.
problem Optimal synthesis in a radially symmetric Grushin space with a weight function.
method Analysis of the geometry of with a weighted Carnot-Carathéodory metric, providing conditions for Grushin-like structure, and describing optimal synthesis.
result Sufficient conditions on the weight function ensure a Grushin-like structure, and the candidate cut time coincides with the true cut time in the integrable case.
Max-Cut decision tree improves classification accuracy and reduces computation time.
problem Improving decision tree accuracy and efficiency for complex classification tasks.
method Alternative splitting metric (max cut) and PCA-based feature selection at each node.
result 49% improvement in accuracy with 94% reduction in CPU time on CIFAR-100 data.
New polynomial-time solutions found for training ReLU networks, mirroring Max-Cut complexity.
problem Training two-layer ReLU neural networks with weight decay regularization.
method Developed a convex formulation and randomized algorithm to find approximate global optimizers.
result First polynomial-time approximation guarantees and hardness of approximation results for regularized ReLU networks.
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest paths on Lie groups with sub-Riemannian metrics.
method Analytical and geometric methods to find geodesics and shortest arcs.
result Found geodesics, shortest arcs, distances, and conjugate loci for specified metrics.
This work reviews left-invariant optimal control problems on Lie groups.
problem Optimal control problems on Lie groups with big symmetry.
method Review of main notions, methods, and results.
result Description of extremal trajectories and their optimality, cut time and cut locus, optimal synthesis.
We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the …
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest arcs in sub-Riemannian metrics on Lie groups.
method Investigated left-invariant sub-Riemannian metrics on and .
result Found geodesics, shortest arcs, cut loci, and conjugate loci.
This article deals with 2d almost Riemannian structures, which are generalized Riemannian structures on manifolds of dimension 2. Such sub-Riemannian structures can be locally defined by a pair of vector fields (X,Y), playing the role of orthonormal frame, that may become colinear on some subset. We denote D = span(X,Y…
This research introduces dynamic portfolio cuts using a spectral approach for graph-theoretic diversification.
problem Traditional methods for estimating asset-return covariance assume statistical time-invariance, failing to capture the nonstationary nature of asset price movements.
method Introduces graph spectral estimators that account for nonstationarity, partitioning the market graph into time-evolving clusters for dynamic portfolio cuts.
result Demonstrates the advantages of the proposed framework over traditional methods through numerical case studies using real-world price data.
The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic nonholonomic systems in four-dimensional space with two-dimensional control, for instance to a system which describes motion of mobile robot with a trailer. The global optimality of extre…
An Improved Cutting Plane Method for Convex Optimization, Convex-Concave Games and its Applicationscs.DS
Improved cutting plane method for convex optimization and games.
problem Efficiently finding points in convex sets or proving they do not contain balls.
method Optimal cutting plane algorithm using leverage scores and advanced data structures.
result Significant improvement in time complexity for convex optimization and games.
The abstract constructs a set of bad 3-orbifolds and shows how any bad 3-orbifold can be transformed into a good one.
problem Characterizing and transforming bad 3-orbifolds into good ones.
method Explicit construction of bad 3-orbifolds and a method of cutting-and-capping to transform them.
result Any bad 3-orbifold can be transformed into a good 3-orbifold through a finite number of operations.
Spectral clustering has become one of the most widely used clustering techniques when the structure of the individual clusters is non-convex or highly anisotropic. Yet, despite its immense popularity, there exists fairly little theory about performance guarantees for spectral clustering. This issue is partly due to the…
The study identifies conjugate and cut points in ideal fluid motion configurations.
problem Understanding stability and re-convergence of fluid configurations.
method Existence and non-existence of conjugate points in specific fluid configurations, using geometric and physical analysis.
result Existence of conjugate points in Kolmogorov flows and non-existence in Arnold steady states.
NeuralCut learns to select cutting planes by looking ahead, outperforming traditional methods.
problem Selecting effective cutting planes for MILP optimization.
method Imitation learning on a lookahead expert to train a neural network for cut selection.
result NeuralCut outperforms standard baselines in cut selection for MILP benchmarks.
NeVI-Cut uses neural networks to efficiently propagate uncertainty without feedback.
problem Efficiently propagating uncertainty in downstream Bayesian analysis without feedback.
method NeVI-Cut combines neural networks and normalizing flows for variational inference.
result NeVI-Cut achieves significant computational gains and higher accuracy than traditional methods.
Machining processes are most accurately described using complex dynamical systems that include nonlinearities, time delays, and stochastic effects. Due to the nature of these models as well as the practical challenges which include time-varying parameters, the transition from numerical/analytical modeling of machining …
The paper connects cut locus, Thom space, and Morse-Bott functions in Riemannian geometry.
problem Analyzing the square of the distance function to a submanifold in a Riemannian manifold.
method Investigates the Morse-Bott property of the square of the distance function on the complement of the cut locus.
result The Thom space of the normal bundle of a submanifold is homeomorphic to the quotient space of the complement of the cut locus.
Paper connects probability density cuts to graph theory eigenfunctions.
problem Developing sparse cuts for probability densities.
method Defines sparse cuts and principal eigenfunctions for probability densities, proving Cheeger and Buser inequalities.
result No such inequalities hold for prior definitions, proving new inequalities for probability densities.
Distance from a Finsler Submanifold to its Cut Locus and the Existence of a Tubular Neighborhoodmath.DG
The paper proves the existence of a tubular neighborhood for Finsler submanifolds.
problem Existence of a tubular neighborhood for Finsler submanifolds.
method Geometric proof of the existence of a tubular neighborhood for Finsler submanifolds.
result The distance between a Finsler submanifold and its cut locus is at least ε when the submanifold is compact.
New equivalence relation for links using cut-diagrams.
problem Classical link concordance.
method Cut-diagrams and cut-concordance.
result Nilpotent peripheral system invariant of cut-concordance.
New algorithm clusters Gaussian mixtures with unknown covariance efficiently.
problem Clustering data from a mixture of Gaussians with unknown covariance.
method Developed an efficient spectral algorithm based on a Max-Cut integer program.
result Achieves optimal misclassification rate with quadratic sample size.
From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data cloudsmath.SP
Study shows convergence rates for Cheeger cuts on data clouds.
problem Optimizing graph cuts for clustering data sampled from a manifold.
method Analyzes statistical properties of Cheeger cuts on proximity graphs built from data.
result Obtains high probability convergence rates for Cheeger constant and cuts.
Unified framework for differentiable graph partitioning with probabilistic cuts.
problem Lack of general guarantees and principled gradients in prior probabilistic relaxations of graph cuts.
method Unified probabilistic framework covering a wide class of cuts, including Normalized Cut, with tight analytic upper bounds.
result Rigorous, numerically stable foundation for scalable, differentiable graph partitioning.
Stochastic cutting planes improve data-driven optimization speed.
problem Data-driven Mixed-Integer Nonlinear Optimization problems.
method Stochastic version of cutting-plane method.
result Stochastic algorithm converges to ε-optimal solution with high probability.
Study of Randers metrics on spheres with simple cut loci.
problem Understanding Randers metrics on spheres and their cut loci.
method Analyzing geodesics, conjugate, and cut loci of Finsler metrics of Randers type.
result Found new families of Randers metrics with simple cut loci.
We define spin-c prequantization of a symplectic manifold to be a spin-c structure and a connection which are compatible with the symplectic form. We describe the cutting of an S^1-equivariant spin-c prequantization. The cutting process involves a choice of a spin-c prequantization for the complex plane. We prove that …
Kaehler cutsmath.DG
A symplectic cut of a manifold M with a Hamiltonian circle action is a symplectic quotient of M x C. If M is Kaehler then, since C is Kaehler, the cut space is Kaehler as well. The symplectic structure on the cut is well understood. In this paper we describe the complex structure (and hence the metric) on the cut. We t…
Max flow/min cut theorem extended to currents and topology.
problem Continuous max flow/min cut theorem for complex domains.
method Continuous analogue of max flow/min cut theorem considering topology.
result Continuous max flow/min cut theorem proven for currents and laminations.
Wu has shown that if a link or a knot in in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that is prime, then the thin sphere of lowest width also does not have any vertical c…
Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced -cut of the graph, are either based on greedy techniques or heuristics which have weak connection to th…
Spectral clustering is sensitive to how graphs are constructed from data particularly when proximal and imbalanced clusters are present. We show that Ratio-Cut (RCut) or normalized cut (NCut) objectives are not tailored to imbalanced data since they tend to emphasize cut sizes over cut values. We propose a graph partit…
The paper studies the cut locus of submanifolds in Riemannian manifolds, providing geometric and topological insights.
problem Understanding the cut locus of submanifolds in Riemannian geometry.
method Analyzing the square of the distance function and using gradient flow lines to deform spaces.
result The cut locus of a submanifold is invariant under certain group actions and provides a deformation retraction.
New algorithm solves large cardinality-constrained clustering problems.
problem Optimizing clustering with cardinality constraints.
method Branch-and-cut technique with SDP relaxation and polyhedral cuts.
result Solves real-world instances 10 times larger than previous methods.