Study relaxed curvature for surfaces, focusing on energy and BV properties.
problem Defining curvature for non-parametric surfaces with BV and measure properties.
method Examined inscribed polyhedral surfaces to approximate relaxed energy, analyzed BV properties and total curvature.
result Properties of functions with finite relaxed energy, analyzed Schwarz-Peano counterexample.
Paper relaxes convexity assumptions in mean curvature flow results.
problem Relaxing convexity assumptions in mean curvature flow results.
method Proves a generalized Harnack inequality and uses maximum principle.
result Characterizes family of shrinking spheres for ancient solutions.
This note relaxes conditions for Kähler metrics with bounded entropy and scalar curvature.
problem Boundedness conditions for Kähler metrics with bounded entropy and scalar curvature.
method Slightly relaxes the boundedness condition on the scalar curvature.
result Apriori estimates and C3,α estimate for the potential of the Kähler metrics under relaxed conditions. Paper finds maximum curvature of Bézier-spline curves.
problem Finding maximum curvature of Bézier-spline curves.
method Modified B-spline solutions for inverse interpolation problem.
result Determined maximum curvature of Bézier-spline curves.
Study introduces weak elastic energy for curves on Riemannian surfaces.
problem Detecting curvature of curves on Riemannian surfaces.
method Relaxation starting from inscribed geodesic polygonals, defined in normalized isothermal coordinates.
result Relaxed energy detects intrinsic second-order Sobolev regularity and agrees with geodesic curvature.
Survey on scalar curvature stability and related questions.
problem Understanding scalar curvature stability and rigidity phenomena.
method Survey and discussion of existing tools and questions.
result Survey of known results and open questions in scalar curvature stability.
Defines weak normals for irregular curves in high-dimensional spaces.
problem Dealing with irregular curves in high-dimensional Euclidean spaces.
method Using sequences of inscribed polygonals and Gram-Schmidt procedure, introduces a relaxed notion of weak normals.
result Weak normals for irregular curves are the strong limit of approximating polygonals and agree with relaxed energy.
For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For H2-regular graphs we show that bounds for the Willmore energy imply area and diameter bounds. We then consider the L1-lower s…
The ACS criterion is verified for specific hypersurfaces in unit spheres.
problem Verifying the ACS criterion for minimal isoparametric hypersurfaces in unit spheres.
method Moment-relaxation technique and explicit extremal configurations.
result The ACS condition holds under specific conditions on principal curvatures.
The paper studies ideal flows of closed curves, classifying critical points and proving flow behavior.
problem Analyzing the generalised ideal flow of closed planar curves.
method Completely classifies critical points and proves properties of the m-ideal flow. result For m>1, the m-ideal flow of closed curves converges to a round multiply-covered circle. In this paper, we consider the classical variational problem in the Galilean space. we develop the Euler-Lagrange equations for a elastic line on an oriented surface in the Galilean 3-dimensional space G3. Using the varia- tion method, we will try to give some characterization for the solution curve (the elastic lin…
Proves existence of sphere foliations with prescribed mean curvature on Riemannian manifolds.
problem Finding sphere foliations with prescribed mean curvature on Riemannian manifolds.
method Proves existence of foliations by spheres with mean curvature proportional to a given function on non-degenerate critical points.
result Essentially unique foliation of spheres with prescribed mean curvature exists in a neighborhood of a non-degenerate critical point.
Study curvature operators with sectional curvature bounds using convex algebraic geometry.
problem Characterize algebraic curvature operators with sectional curvature bounds.
method Apply convex algebraic geometry techniques, including spectrahedra and hierarchies of inner and outer approximations.
result For n≥5, the set of curvature operators is a spectrahedron or a spectrahedral shadow, providing counter-examples to the Helton--Nie Conjecture. A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S^4 or diffeomorphic to …
It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, chara…
As first noted in Korevaar, Kusner and Solomon ("KKS"), constant mean curvature implies a homological conservation law for hypersurfaces in ambient spaces with Killing fields.In Theorem 3.5 here, we generalize that law by relaxing the topological restrictions assumed in [KKS] and by allowing a weighted mean curvature f…
New approach to nematic fields on surfaces, relaxing uniformity to quasi-uniformity.
problem Identifying least distorted nematic fields on generic surfaces.
method Relaxing the notion of uniformity into quasi-uniformity and proving parallel transport by geodesics.
result All quasi-uniform fields are parallel transported by the geodesics of the surface.
Paper extends noncompact Llarull's theorem to manifolds with boundary.
problem Finding upper bounds for scalar curvature infimum in noncompact manifolds.
method Using deformed Dirac operators to relax boundary conditions.
result Upper bound for scalar curvature infimum in terms of Laplacian spectrum.
We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal…
The paper proves manifold rigidity under curvature conditions.
problem Rigidity of manifolds with harmonic curvature and curvature operator positivity.
method Analyzes conditions on complete manifolds to prove constant sectional curvature.
result Rigidity holds for manifolds with harmonic curvature and curvature operator positivity.
The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.
problem Determining when a submanifold lies on a shear-free null hypersurface under integral curvature conditions.
method Using Minkowski formulas with arbitrary weight to derive rigidity results for submanifolds with weaker integral curvature conditions.
result A necessary and sufficient condition for a submanifold to lie in a shear-free null hypersurface is given by a mean curvature integral inequality.
Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.
problem Hopf's conjecture on positive sectional curvature and its failure under relaxed conditions.
method Non-trivial extension of the Four Periodicity Theorem to higher degrees.
result Proves positive Euler characteristic for specific manifolds with positive second intermediate Ricci curvature.
Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.
problem Obstructing the existence of positive scalar curvature metrics with mean convex boundaries.
method Atiyah-Patodi-Singer index formula, deformation principle, homotopy equivalences, higher homotopy groups.
result Construction of compact manifolds with nontrivial higher homotopy groups for positive scalar curvature metrics with mean convex boundaries.
KF-LAX uses KFAC to improve sample efficiency in reinforcement learning.
problem Sample efficiency and low variance in gradient-based optimization methods.
method Kronecker-factored curvature estimation (KFAC) applied to RELAX gradient estimator.
result Improved performance on synthetic and Atari games.
Ancient flows of elliptic functionals classified in various dimensions.
problem Classifying ancient solutions to gradient flows of elliptic functionals.
method Analyzing closed ancient solutions in Riemannian manifolds.
result Ancient solutions classified in multiple dimensions and cases.
The paper proves nonexistence results for translating solitons in r-mean curvature flow.
problem Proving nonexistence of translating solitons in r-mean curvature flow.
method Establishing nonexistence results under suitable growth conditions on curvature and second fundamental form.
result Properly immersed translating solitons cannot be confined to certain half-spaces.
An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature, a similar but less explicit formula is also found. Here it is proven that any positive ancient solution is the standard Laplace transform of…
Simplified Obata-Vétois argument for Einstein manifolds with nonnegative scalar curvature.
problem Identifying conditions for closed conformally Einstein manifolds to be Einstein.
method Simplified Obata-Vétois argument, identifying a closed interval containing zero.
result Closed conformally Einstein manifolds with nonnegative scalar curvature are Einstein if they satisfy certain conditions.
Convex relaxations improve CNNs with fixed weights.
problem Improving CNNs with fixed weights.
method Convex relaxations for CNNs with fixed weights using second order cone programs.
result The relaxation recovers the global minimum under a planted model assumption.
New method improves neural network verification by considering multivariate input space of ReLU neurons.
problem Improving the effectiveness of neural network verification algorithms.
method A new tightened convex relaxation for ReLU neurons considering multivariate input space.
result Our convex relaxation is significantly stronger than the commonly used univariate-input relaxation.
New semidefinite relaxation improves robustness certification of neural networks.
problem Certifying robustness of neural networks against adversarial examples.
method Proposed a new semidefinite relaxation for certifying robustness of arbitrary ReLU networks.
result Our proposed relaxation is tighter than previous relaxations and produces meaningful robustness guarantees.
Bipartite graphs with more edges than a threshold have positive curvature.
problem Determining the curvature of bipartite graphs based on edge density.
method Using a new formula for Lin--Lu--Yau curvature, the study establishes conditions for bipartite graphs to have positive curvature.
result Bipartite graphs with more edges than the specified threshold have positive Lin--Lu--Yau curvature.
New regularizers tighten convex relaxation bounds for neural networks.
problem Large gap between certifiable and empirical robustness in neural networks.
method Two regularizers to train neural networks yielding tighter convex relaxation bounds.
result Higher certified accuracy with proposed regularizers.
We study manifolds satisfying a weighed Poincare inequality, which was first introduced by Li-Wang. We generalized one of their results by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and established a finitely many ends result. We proved a vanishing result for L2 harmonic …
Improved neural network robustness certification through tighter convex relaxations.
problem Certifying neural network robustness to perturbed and adversarial inputs.
method Exploiting ReLU network structure, novel partition-based certification procedure.
result Tightens existing linear programming relaxations to achieve zero relaxation error asymptotically.
In this note we compare two recently proposed semidefinite relaxations for the sparse linear regression problem by Pilanci, Wainwright and El Ghaoui (Sparse learning via boolean relaxations, 2015) and Dong, Chen and Linderoth (Relaxation vs. Regularization A conic optimization perspective of statistical variable select…
Paper refines Einstein manifold result with cone curvature condition.
problem Closed Einstein manifolds with specific curvature conditions.
method Relaxing curvature condition to cone condition and proving manifold properties.
result Closed Einstein manifolds of dimension 4, 5, or ≥8 are either flat or round spheres under the cone curvature condition.
The study proves splitting theorems for manifolds with specific curvature and boundary conditions.
problem Proving splitting theorems for manifolds with specific curvature and boundary conditions.
method Warped product splitting theorem in manifolds with Ricci curvature bounded from below, requiring parabolic and convex boundary.
result Established splitting results for various manifolds with specific curvature and boundary conditions.
The study examines spectral rigidity in Ricci solitons and Einstein-type manifolds.
problem Determining sectional curvature from eigenvalues of the p-Laplacian.
method Analyzes spectral rigidity under gradient shrinking Ricci soliton and cohomologically Einstein conditions.
result With some exceptions, sectional curvature can be determined by eigenvalues of the p-Laplacian.
Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. Particularly, we con…
A new Wasserstein K-means method for clustering probability distributions.
problem Clustering probability distributions using the Wasserstein metric.
method Distance-based K-means with SDP relaxation for Wasserstein barycenters. result Distance-based K-means outperforms centroid-based K-means for clustering probability distributions. New elastic energy for irregular curves defined through polygonal approximations.
problem Defining elastic energy for irregular curves in any space dimension.
method Relaxation process with p-rotation of inscribed polygonals, focusing on geometric curvature distribution. result Energy finite if and only if curve's arc-length parameterization has second order summability.
Introduces optimization geometrodynamics for dynamic geometric optimization.
problem Gradient-based optimization methods struggle with changing geometric constraints.
method Optimization geometrodynamics separates invariant and improvable geometric mismatches.
result Dynamic geometric complexity measures the minimum geometric cost to reduce optimization difficulty.
In this work we study convex relaxations of quadratic optimisation problems over permutation matrices. While existing semidefinite programming approaches can achieve remarkably tight relaxations, they have the strong disadvantage that they lift the original n×n-dimensional variable to an n2×n2-d…
Statistical image reconstruction (SIR) methods are studied extensively for X-ray computed tomography (CT) due to the potential of acquiring CT scans with reduced X-ray dose while maintaining image quality. However, the longer reconstruction time of SIR methods hinders their use in X-ray CT in practice. To accelerate st…
The relaxed maximum entropy problem is concerned with finding a probability distribution on a finite set that minimizes the relative entropy to a given prior distribution, while satisfying relaxed max-norm constraints with respect to a third observed multinomial distribution. We study the entire relaxation path for thi…
A new framework describes dissipation using a metriplectic 4-bracket.
problem Describing dissipation in a way that preserves energy and entropy.
method Using a metriplectic 4-bracket, a quantity like the Poisson bracket with symmetries motivated by Riemannian curvature.
result The metriplectic 4-bracket dynamics includes all known previous binary bracket theories for dissipation.
Bayesian learning is often hampered by large computational expense. As a powerful generalization of popular belief propagation, expectation propagation (EP) efficiently approximates the exact Bayesian computation. Nevertheless, EP can be sensitive to outliers and suffer from divergence for difficult cases. To address t…