Asymmetric expansion preserves convexity in hyperbolic geometry.
problem Maintaining convexity in hyperbolic geometry under asymmetric expansions.
method Generalizing earlier results on radial expansion to asymmetric expansion.
result Asymmetric expansion of hyperbolic convex sets remains convex.
Simplified proof of Honda-Huang's contact convexity result.
problem Contact convexity in high dimensions
method Simplified proof of Honda-Huang's main result
result Main result from Honda-Huang's paper simplified and presented
Let U⊆Rn be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function f:U→R can be approximated by real analytic convex functions, uniformly on all of U. In doing so we provide a technique which transfers results on uniform approximation on bounded …
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C1,1 functions. Local rigidity proved for convex hypersurfaces in spaces of constant curvature.
problem Proving rigidity of convex hypersurfaces in spaces of constant curvature.
method Analyzing isometric convex hypersurfaces in spaces of constant curvature of dimension n≥4. result Two convex isometric hypersurfaces are congruent locally around their corresponding under strict convexity isometries.
As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic φ-convex function and deduce some basic properties of φ-convex function and geodesic φ-convex function. We also introduce the concept of geodesic φ-convex set and φ-epigraph and in…
The paper studies how convex surfaces shrink under mean curvature flow with a free boundary.
problem Mean curvature flow of convex surfaces with a free boundary on convex barriers.
method Introduced a new perturbation argument to establish convexity and pinching estimates.
result The flow contracts a sufficiently convex surface to a point in finite time, asymptotic to a half-sphere.
Proves flows of two-convex Lagrangians are regular, global, and converge.
problem Proves regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case.
method Uses a newly discovered monotone quantity to control two-convexity.
result Proves results for the mean curvature flow of area-decreasing Lagrangian submanifolds.
The paper proves convexity of certain solitons and expanders in high dimensions.
problem Proving convexity of specific solitons and expanders in Rn+1. method Inspired by Spruck-Xiao and Derdziński, the paper uses geometric analysis to prove convexity.
result The paper proves the convexity of complete 2-convex translating and expanding solitons and expanders in Rn+1 for n≥3. New convexity concept applied to sphere yields quermassintegral inequalities.
problem Proving quermassintegral inequalities for horo-convex hypersurfaces on the sphere.
method Smooth convergence of Guan/Li flow for inverse type applied to horo-convex hypersurfaces.
result Full set of quermassintegral inequalities for horo-convex hypersurfaces proved.
In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension 2n+2 which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on ∂W, which we call \emph{convex open book}, induced b…
Generalizes rigidity of scalar curvature for convex domains.
problem Rigidity of scalar curvature for convex domains.
method Harmonic spinors on convex domains with boundary conditions constructed by Brendle.
result Rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain.
We obtain upper and lower bounds on the difference between the renormalized volume and the volume of the convex core of a convex cocompact hyperbolic 3-manifold which depend on the injectivity radius of the boundary of the universal cover of the convex core and the Euler characteristic of the boundary of the convex cor…
Improved kernel quadrature with convex weights using subsampling.
problem Constructing quadrature rules with small worst-case error.
method Combining spectral properties of the kernel with recombination results.
result Effective algorithms for constructing convex quadrature rules with i.i.d. samples.
In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class C1. Moreover, applications of this result are given.
Study cash-subadditive risk measures without quasi-convexity.
problem Cash subadditivity without quasi-convexity.
method Represent cash-subadditive risk measures as lower envelopes of quasi-convex measures and introduce quasi-star-shapedness.
result General cash-subadditive risk measures can be represented as lower envelopes of quasi-convex measures.
Wang simplifies ancient convex solutions to mean curvature flow structure theory.
problem Understanding convex ancient solutions to mean curvature flow.
method Simplified analysis using monotonicity formula and differential Harnack inequality, new structure result.
result Various rigidity results for convex ancient solutions and translators follow from structure theory.
Three results in p-convex geometry are established. First is the analogue of the Levi problem in several complex variables, namely: local p-convexity implies global p-convexity. The second asserts that the support of a minimal p-dimensional current is contained in the p-hull of the boundary union with the "core" of the…
Convex clustering can only learn convex clusters, with significant gaps between clusters.
problem Understanding the limitations and capabilities of convex clustering.
method Analyzing convex clustering solutions, proving properties, and characterizing clusters.
result Convex clustering can only learn convex clusters with significant gaps between clusters.
In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any …
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.
Proves convexity of minimizers in energy functions with convex potentials.
problem Connectedness and convexity of minimizers in energy functions involving surface tensions and convex potentials.
method Introduces a 'two-point function' to measure lack of convexity and prove negative second variation of the energy.
result Positively answers an old question of Almgren about connectedness and convexity of minimizers.
No non-trivial convex functions on certain Finsler manifolds.
problem Existence of convex functions on Finsler manifolds.
method Analyzing Holmes- Thompson volume and Busemann function properties.
result Non-compact Finsler manifolds with infinite Holmes- Thompson volume have no non-trivial convex functions.
Least Squares Estimators are suboptimal for 5D convex functions.
problem Suboptimality of Least Squares Estimators in estimating multidimensional convex functions.
method Analysis of natural subclasses of convex functions in random and fixed design settings.
result Risk of LSE is n−2/d while minimax risk is n−4/(d+4) for d≥5. Sharp estimates for Finsler metrics in convex domains.
problem Estimating distances in Finsler metrics near convex points.
method Sharp estimates for intrinsic distances of Finsler metrics.
result Characterization of k-quasi hyperbolic metric in convex geometry. Optimal risk sharing without convex preferences using aggregate convexity.
problem Risk sharing among non-convex preferences.
method Aggregate convexity principles and Lyapunov convexity, combined with approximation arguments for law invariant risk measures.
result Derivation of a computationally tractable formula for the conjugate of the value function.
Analyzes singularities of convex hypersurfaces in hyperbolic space.
problem Understanding singularities of convex hypersurfaces with constant curvature.
method Analyzes the structure of singular sets using convex curvature functions.
result Describes the structure of singular sets in hyperbolic space.
The paper proves convexity results for a specific type of Lie groups.
problem Convexity results for non-compact real reductive Lie groups.
method Proves convexity results through orbit projection.
result Convexity results for quasi-hermitian Lie groups.
In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polyn…
Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
New guarantees for Group LASSO in sparse convex optimization.
problem Sparse convex optimization with vector-valued features.
method Group LASSO regularization and analysis of gradient norms.
result Group LASSO selects the same features as Orthogonal Matching Pursuit.
Study limits of convex domains in projective plane, proving specific results.
problem Understanding limits of convex domains in projective plane.
method Analyzing sequences of properly convex domains with bounded multiplicity.
result Determined all Hausdorff limit domains after normalization.
The paper shows that g-convex functions on manifolds are sparse.
problem Characterizing and understanding the sparseness of g-convex functions.
method Established criteria for g-convexity and used them to prove sparseness results.
result Most g-convex functions on compact manifolds have few critical points.
Random walks generate quasi-convex subgroups in acylindrically hyperbolic groups.
problem Understanding the structure of subgroups generated by random walks in hyperbolic groups.
method Probabilistic approach using random walks in acylindrically hyperbolic groups.
result Subgroups generated by random walks are quasi-convex and isomorphic to the original subgroup and the random walk subgroup.
Paper uses integer programming for non-convex boosting in classification.
problem Improving classification performance using non-convex optimization.
method Non-convex boosting via integer programming.
result Results comparable to or better than state-of-the-art.
In this paper we show that bending a finite volume hyperbolic d-manifold M along a totally geodesic hypersurface Σ results in a properly convex projective structure on M with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We th…
Paper proves rigidity of convex hypersurfaces in various spaces.
problem Proving the uniqueness of convex hypersurfaces in multidimensional spaces.
method Generalizing Senkin's theorem to higher dimensions and constant curvature spaces.
result Rigidity of convex hypersurfaces in En+1, n≥3. Generalizes Giroux's result to higher dimensions and applies to contact manifolds.
problem Characterizing convex surfaces in contact manifolds.
method Extending Giroux's result to arbitrary dimensions and applying to specific hypersurfaces.
result Closed hypersurfaces are C∞-close to convex hypersurfaces. Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.
Convex optimization with sparsity-promoting convex regularization is a standard approach for estimating sparse signals in noise. In order to promote sparsity more strongly than convex regularization, it is also standard practice to employ non-convex optimization. In this paper, we take a third approach. We utilize a no…
Greedy convex combination of models outperforms baselines.
problem Overfitting and underfitting in convex combinations of models.
method Greedy learning of convex combinations with early stopping.
result Greedy approach is competitive or better than boosting and random forests.
New methods for convex optimization with locally Lipschitz gradient, achieving faster convergence.
problem Optimization problems with locally Lipschitz continuous gradient.
method Accelerated proximal gradient (APG) methods and proximal augmented Lagrangian method.
result Achieved faster convergence rates for convex optimization problems with locally Lipschitz gradient.
Affine deformations of convex cones on projective surfaces.
problem Understanding affine actions on convex cones.
method Geometric correspondence and convex tube domains.
result Quotients of convex domains are affine manifolds with convex surfaces.
This paper improves total variation based convex clustering for better data clustering.
problem Improving data clustering methods, especially for general data.
method Proposes a weighted sum-of-ℓ1-norm relating convex model for total variation based clustering. result Established exact clustering property applicable to general data, sharper than existing results.
New capillary Christoffel-Minkowski problem solved for half-space.
problem Finding strictly convex capillary hypersurfaces from capillary functions.
method Established analogous result in capillary setting for half-space.
result Capillary functions lead to strictly convex capillary hypersurfaces.
CoNES optimizes blackbox functions using convex optimization and information geometry.
problem Optimizing high-dimensional blackbox functions efficiently.
method Formulated as a convex program that adapts evolutionary strategies gradient estimates.
result Vastly outperforms conventional blackbox optimization methods on benchmarks and MuJoCo tasks.
Optimally shows the distance between perturbed convex functions and their Γ-regularizations.
problem Understanding the difference between perturbed convex functions and their Γ-regularizations.
method Analyzing the compactly supported perturbation and the Γ-regularization of a strictly convex function.
result The optimal estimate of the distance between perturbed convex functions and their Γ-regularizations is shown to be o(ε).