An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have boundary, and have word hyperbolic divid…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
Trend · papers per month
48 results for “complex convex sets”
The paper examines conditions for the Kobayashi metric to be Gromov hyperbolic on complex convex sets.
problem Conditions for the Kobayashi metric to be Gromov hyperbolic on complex convex sets.
method Develops necessary and sufficient conditions for Gromov hyperbolicity, extends results to -smooth boundaries, and provides counterexamples.
result Gromov hyperbolicity of Kobayashi metric on -convex sets with -smooth boundary, and counterexamples showing boundary regularity is necessary.
New minimal surfaces found with Cantor ends in convex domains.
problem Finding complex structures for minimal surfaces with Cantor ends.
method Proving existence of complete minimal surfaces with Cantor ends in minimally convex domains.
result Existence of a Cantor set whose complement forms a complete minimal surface.
The paper proves a rigidity theorem for non-compact convex sets in hyperbolic 3-space.
problem Determining a closed convex set in hyperbolic 3-space by its boundary metric.
method Pogorelov's rigidity theorem, Hausdorff measure, and complex analysis techniques.
result The intrinsic path metric on the boundary determines a closed convex set up to isometry under certain conditions.
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.
Paper improves privacy in ERM with faster algorithms and broader applicability.
problem Privacy-preserving machine learning with empirical risk minimization.
method Develops faster algorithms for differentially private ERM in various settings.
result Achieves optimal or near-optimal utility bounds with less gradient complexity.
Develops theory of relatively geometric actions on CAT(0) cube complexes.
problem Understand actions of relatively hyperbolic groups on CAT(0) cube complexes.
method Introduces and studies relatively geometric actions, proving key results.
result Proves full relatively quasi-convex subgroups are convex compact.
The paper proves conditions for convex domains to be strongly pseudoconvex.
problem Conditions for convex domains to be strongly pseudoconvex.
method Establishes gap theorem for complex geometry of convex domains.
result Conditions for convex domains to be strongly pseudoconvex.
Adaptive exploration scheme for evaluating multiple policies with different rewards.
problem Online multi-reward multi-policy evaluation.
method Adapted -PAC perspective and MR-NaS exploration scheme to minimize sample complexity.
result Demonstrated effectiveness of adaptive exploration in tabular domains.
New research shows parallel optimization is ineffective for convex problems.
problem The inefficiency of parallel optimization methods for convex problems.
method Lower bounds analysis in the local oracle model of computation.
result Parallel and randomized algorithms cannot speed up convex optimization in various geometries and objective functions.
Improved variance reduction for Riemannian non-convex optimization with adaptive batch size.
problem Optimizing non-convex functions on Riemannian manifolds.
method Batch size adaptation in R-SVRG, R-SRG, and R-SPIDER.
result Achieves lower total complexities for various non-convex functions.
New method solves complex optimization problems faster.
problem Minimizing a convex smooth objective over the optimal solution set of another convex smooth problem.
method Uses a cutting plane approach to approximate the lower-level problem and an accelerated gradient method to update the upper-level objective.
result Shows that the method requires at most iterations to achieve -suboptimality and -infeasibility.
This paper studies quasar-convex functions to improve optimization methods.
problem Improving optimization methods for non-convex functions.
method Study of first order methods for quasar-convex functions.
result Proves complexity upper bounds similar to convex functions.
Optimal Guarantees for Algorithmic Reproducibility and Gradient Complexity in Convex Optimizationcs.LG
New algorithms ensure reproducibility and optimal convergence in convex optimization.
problem Trade-off between reproducibility and convergence rate in convex optimization.
method Regularization-based algorithms for smooth convex minimization and minimax optimization.
result Achieves optimal reproducibility and near-optimal gradient complexity for various oracle settings.
New methods solve complex optimization problems without strong convexity assumptions.
problem Complex bilevel optimization problems with minimax lower-level structures.
method Penalty-based first-order methods for bilevel minimax optimization.
result Achieves -KKT point with improved oracle complexity.
For a strongly pseudo-convex complex Finsler manifold M, a bundle U of adapted unitary frames is canonically defined. A non-linear Hermitian connection on U, invariant under local biholomorphic isometries, is given and it proved to be unique. By means of such connection, an absolute parallelism on U is determined and a…
New algorithms improve agnostic learning for triangles and polygons, reducing time complexity.
problem Efficient agnostic learning for geometric concept classes.
method Data structures and algorithms from computational geometry, probabilistic combinatorics.
result Optimal time complexity improvements for agnostic learning of triangles and polygons.
The paper proves abelian convexity theorems using Kempf-Ness functions.
problem Establishing convexity along orbits in general settings.
method Using Kempf-Ness functions to prove abelian convexity theorems.
result Short proofs of Atiyah-Guillemin-Sternberg theorem and abelian convexity for gradient maps.
New algorithms optimize non-smooth, non-convex objectives with improved complexity.
problem Optimizing non-smooth, non-convex stochastic objectives.
method Reduction to online learning, applying optimistic online learning techniques.
result Improved complexity for finding -stationary points.
New findings show Rademacher complexities are not crucial for learning complexities.
problem Understanding the sample complexity of learning with squared loss in convex classes.
method Novel learning procedure combining mean estimation and Talagrand's generic chaining method.
result Sample complexity is determined by the limiting Gaussian process, not Rademacher complexities.
Optimizes average of convex functions with tight bounds.
problem Minimizing the average of m convex functions with gradient and prox oracles.
method Tight upper and lower bounds on complexity for deterministic and randomized optimization.
result Optimal methods for smooth and non-smooth functions, showing significant gap between deterministic and randomized settings.
A generalized optimistic method for saddle point problems with improved complexity.
problem Solving convex-concave saddle point problems efficiently.
method Proposes a generalized optimistic method that includes the optimistic gradient method as a special case, handling constrained saddle point problems with composite objective functions and arbitrary norms.
result Best-known global iteration complexity bounds for first-, second-, and higher-order methods.
Paper solves a complex equation for unbounded convex sets.
problem Solving the dual Minkowski problem for unbounded closed sets.
method Using variational properties of Monge-Ampère functionals, the paper proves existence, regularity, and uniqueness of solutions.
result Existence, regularity, and uniqueness of solutions to the Monge-Ampère type equation for .
Constructs hyperbolic reflection groups with 3D limit sets.
problem Existence of convex cocompact groups with specific limit sets.
method Inputting a simplicial complex into a construction process yields a hyperbolic reflection group.
result Answers Kapovich's question affirmatively by creating a thin subgroup of an arithmetic lattice.
In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the regular set can be joined by a smooth minimal geodesic lying entirely in the regul…
Projection-Free Methods for Stochastic Simple Bilevel Optimization with Convex Lower-level Problemmath.OC
New methods optimize complex optimization problems with improved efficiency.
problem Optimizing complex problems with a convex lower-level objective.
method Uses stochastic cutting planes and conditional gradient updates.
result Improves complexity for both convex and non-convex upper-level functions.
RSG method reduces subgradient method's complexity for convex optimization.
problem Finding optimal solutions for convex optimization problems efficiently.
method Periodically restarts the standard subgradient method to reduce complexity.
result RSG method finds -optimal solutions with lower complexity than SG.
Novel convex surrogate for submodular losses with tractable computation.
problem Learning with non-modular losses for set prediction.
method Proposed Lovász hinge loss function for submodular losses.
result First tractable convex surrogates for submodular losses.
Convex meets complexmath.CV
Survey explores interactions between convex and complex geometry.
problem Understanding intersections between convex and complex geometry.
method Survey and review of existing literature.
result Demonstrates fascinating interactions between convex and complex geometry.
Paper studies how to combine regret minimizers for solving complex games.
problem Solving large-scale extensive-form games with constraints.
method Derives a calculus for constructing regret minimizers for composite convex sets.
result Local regret minimizers for simpler sets can be combined into an aggregate for composite sets.
The paper shows how to simplify complex optimization problems into simpler ones.
problem Complex multiobjective optimization problems.
method Proving strongly convex problems are simplicial under certain conditions and demonstrating transformations.
result Strongly convex problems can be simplified into simpler ones via generic linear perturbations.
We find a convex model for traditional nonlinear regression under L2 loss.
problem Nonlinear regression under L2 loss with non-convex optimization.
method Showed a convex nonlinear regression model for least squares problem.
result Existence of a convex model simplifies training complex systems.
Deep neural networks help recover two signals from noisy mixtures.
problem Recovering two signals from noisy subgaussian mixtures with prior structural information.
method Used deep generative neural networks (GNNs) to solve the demixing problem for Lipschitz signals.
result Proved a sample complexity bound for nearly optimal recovery error, extending previous results.
We focus on the problem of minimizing a convex function over a convex set given queries to a stochastic first order oracle. We argue that the complexity of convex minimization is only determined by the rate of growth of the function around its minimizer , as quantified by a Tsybakov-like noise co…
The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.
problem Understanding the Lee-Gauduchon cone for complex manifolds.
method Analyzing the Lee-Gauduchon cone as a convex cone of cohomology classes.
result The Lee-Gauduchon cone is a bimeromorphic invariant.
New algorithm finds optimal sample complexity for pure exploration with multiple good answers.
problem Determining the optimal number of samples needed to explore multiple good answers in a bandit problem.
method Derive lower bound using game equilibrium, extend Track-and-Stop algorithm to multiple answers.
result New algorithm has asymptotic sample complexity matching the derived lower bound.
For convex domains, automorphism group and limit set properties are described.
problem Characterize the automorphism group and limit set of convex domains.
method Detailed analysis of automorphism group structure and limit set properties for convex domains with boundary.
result The automorphism group has finitely many components and the limit set is homeomorphic to a sphere.
The study of Kähler metrics on domains restricts their boundary geometry.
problem Understanding the geometry of domains with negatively pinched Kähler metrics.
method Analyzing the existence and properties of negatively pinched Kähler metrics on domains.
result The boundary of a convex domain without complex subvarieties of positive domain if it admits a complete Kähler metric with pinched negative holomorphic bisectional curvature.
The curves not carriedmath.GT
Suppose is a train track on a surface . Let be the set of isotopy classes of simple closed curves carried by . Masur and Minsky [2004] prove is quasi-convex inside the curve complex . We prove the complement, , is quasi-convex.
Paper revisits set membership estimation for linear systems with relaxed disturbance bounds.
problem Set membership estimation for linear systems with disturbances bounded by convex sets.
method Adopted block-martingale small-ball condition and random perturbed control policies to establish convergence rates.
result Established convergence rates for disturbances bounded by general convex sets.
Improved Frank-Wolfe algorithm for constrained convex optimization with nearest extreme point oracle.
problem Constrained smooth convex minimization with limited linear optimization oracle access.
method Frank-Wolfe algorithm with nearest extreme point oracle.
result Improved complexity bounds for specific feasible sets, including linear convergence for polytopes.
Optimal algorithm finds if point is in convex hull of distributions.
problem Determining if a point is inside the convex hull of means of multiple distributions.
method Thompson-CHM algorithm with modular design of stopping and sampling rules.
result First asymptotically optimal algorithm for CHM problem in one dimension.
Paper solves minimax optimization gap with near-optimal algorithms.
problem Designing efficient algorithms for smooth and strongly-convex-strongly-concave minimax problems.
method Accelerated proximal point method and accelerated solver for minimax proximal steps.
result First algorithm with gradient complexity matching the lower bound up to logarithmic factors.
Let S be the boundary of a handlebody M. We prove that the set of curves in S that are boundaries of disks in M, considered as a subset of the complex of curves of S, is quasi-convex.
In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics in which are invariant with respect to the natural action of the real torus $(\Bbb S^1)^n…
First-order methods tackle g-convex optimization on Hadamard manifolds.
problem Geodesically convex optimization on nonlinear metric spaces.
method Iteration complexity analysis for first-order algorithms.
result Upper bounds for global complexity of g-convex optimization.
Study quotients of curve complex actions by mapping class group.
problem Understanding actions of mapping class group on curve complex quotients.
method Cone off uniformly quasi-convex subspaces to form symmetric curve sets, non-maximal train track sets, and compression body disc sets. Analyze actions of mapping class group on these quotients.
result Actions of mapping class group on quotients are strongly WPD, non-elementary, and have infinite diameter.
A new method for distributed optimization reduces communication rounds without minibatches.
problem Efficient training in distributed machine learning with different data distributions.
method A primal-dual method (GA-MSGD) applied to the Lagrangian of distributed optimization.
result Achieves linear convergence in communication rounds for strongly convex objectives.