Bound on equivariant index for min-max surfaces.
problem Bounding the index of equivariant min-max surfaces.
method Equivariant min-max procedure with group action.
result Equivariant index bound by number of parameters.
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48 results for “min-max procedure”
We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lie in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Co…
Study proves properties of constant mean curvature hypersurfaces in high-dimensional spaces.
problem Properties of constant mean curvature hypersurfaces in high-dimensional spaces.
method Proves properties of constant mean curvature hypersurfaces using min-max procedure and surgery.
result Every tangent cone at each isolated singularity is area-minimising.
We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. We also construct counter-examples in dimension and to the -regularity in the convergence procedure. Furthermore, we prove the lower semi-continuity of the index o…
Proves existence of special 2-spheres in curved 3-spaces.
problem Existence of constant mean curvature 2-spheres in Riemannian 3-spheres.
method Develops a min-max scheme for a weighted Dirichlet energy functional, using bi-harmonic approximation, derivative estimates, and Morse index estimates.
result Proves existence for almost every mean curvature and all for positively curved 3-spheres.
We use a min-max procedure on the Allen-Cahn energy functional to construct geodesics on closed, 2-dimensional Riemannian manifolds, as motivated by the work of Guaraco. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to Wang-Wei, we manage…
Entropy is a natural geometric quantity measuring the complexity of a surface embedded in . For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface is at least that of the self-shrinking two-sphere. We prove this conjecture …
New minimal surfaces found in 3D sphere with low genus.
problem Finding minimal surfaces in 3D sphere with low genus.
method Equivariant min-max procedure applied to a novel sweepout.
result Discovered new minimal surfaces with low genus.
We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such …
Motivated by applications in Game Theory, Optimization, and Generative Adversarial Networks, recent work of Daskalakis et al \cite{DISZ17} and follow-up work of Liang and Stokes \cite{LiangS18} have established that a variant of the widely used Gradient Descent/Ascent procedure, called "Optimistic Gradient Descent/Asce…
Improves model generalization by minimizing loss sharpness.
problem Overparameterized models often fail to generalize well despite low training loss.
method Sharpness-Aware Minimization (SAM) minimizes both loss value and sharpness.
result SAM improves model generalization across various datasets and models.
Study shows strong min-max principle for phase transitions.
problem Understanding nodal sets near minimal hypersurfaces.
method Analogous to White's principle, applies to Allen-Cahn energy.
result Strong min-max principle for phase transitions.
Equity-Transformer solves NP-hard min-max routing problems efficiently.
problem Min-max routing problems with multiple agents and large-scale applications.
method Sequential planning approach with Transformer and equitable workload distribution inductive biases.
result Significant runtime and cost reductions in min-max mTSP and min-max mPDP tasks.
Upper bound for Morse index of min-max varifolds.
problem Bounding Morse index of varifolds.
method Proving upper bound for Morse index of min-max stationary integral varifolds.
result Upper bound for Morse index of min-max stationary integral varifolds.
Localized min-max method proves minimal hypersurface existence.
problem Existence of minimal hypersurfaces in complete manifolds.
method Localized min-max approach to prove existence.
result Existence of complete embedded minimal hypersurface with index at most one.
The paper solves min-max widths on a 3-sphere and strengthens multiplicity theorems.
problem Which min-max widths of the unit 3-sphere lie between and ?
method Homological min-max theory and stronger versions of multiplicity one theorems.
result Proves the 10th to 13th min-max widths of the unit 3-sphere lie between and .
Study on double descent behavior in two-layer neural networks for binary classification.
problem Understanding the double descent phenomenon in model test error.
method Two-layer neural network with ReLU activation for binary classification. Quantified model size by sample-to-dimension ratio. Empirical risk minimization using Convex Gaussian Min Max Theorem.
result Observed and investigated the double descent behavior of model test error.
Study introduces statistical mechanics for min-max problems.
problem Understanding the properties of min-max problems in high dimensions.
method Statistical mechanical formalism for analyzing min-max problems.
result Derives the relationship between training data and generalization error.
Adaptive momentum method solves non-convex min-max problems.
problem Non-convex min-max optimization problems in training generative adversarial networks.
method Proposes an adaptive momentum algorithm for non-convex min-max optimization.
result Establishes non-asymptotic convergence rates for the proposed algorithm.
Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.
problem Existence and finiteness of -invariant minimal hypersurfaces.
method Equivariant min-max theory, compactness theorem, bumpy metrics theorem.
result Generalization of Morse index estimates to equivariant setting.
New proof of Smale conjecture for RP^3 and lens spaces using min-max theory.
problem Proving the Smale conjecture for specific spaces.
method Minimal surfaces and min-max theory.
result New proof of Smale conjecture for and lens spaces.
Paper improves Morse index bound for hypersurfaces.
problem Improving Morse index bound for hypersurfaces.
method Construction of hierarchical deformations and restrictive min-max theory.
result Generalizes a result by X. Zhou for .
New methods solve min-max problems on manifolds using Riemannian Hamiltonians.
problem Min-max optimization on Riemannian manifolds.
method Riemannian Hamiltonian methods (RHM) to minimize the Hamiltonian function.
result RHM leads to correct search directions and global optimality in min-max problems.
Interpretable transformed ANOVA approximation on the example of the prevention of forest firesstat.ML
The paper uses transformed ANOVA to identify important fire detection variables.
problem Identifying key variables for forest fire detection.
method Developed a complete orthonormal system for standard normal distribution, applied Z-score transformation, and used ANOVA approximation.
result Attribute ranking reveals important variables for fire detection.
We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each…
The combined work of Guaraco, Hutchinson, Tonegawa and Wickramasekera has recently produced a new proof of the classical theorem that any closed Riemannian manifold of dimension contains a minimal hypersurface with a singular set of Hausdorff dimension at most . This proof avoids the Almgren--Pitts …
Survey of advances in non-convex min-max optimization for applications.
problem Finding optimal solutions in non-convex, non-concave min-max problems.
method Selective review of theoretical and algorithmic advances.
result Exciting recent advances in solving non-convex min-max problems.
We prove that for generic metrics on a 3-sphere, the minimal surface obtained from the min-max procedure of Simon-Smith has index 1. We prove an analogous result for minimal surfaces arising from strongly irreducible Heegaard sweepouts in 3-manifolds. We also confirm a conjecture of Pitts-Rubinstein that a strongly irr…
We reformulate LIPs as min-max problems for easier solution.
problem Recovering signals from few linear measurements.
method Proposed a min-max reformulation of LIPs.
result Saddle points characterize solutions to LIPs.
Epoch-GDA achieves optimal convergence rate for SCSC min-max problems.
problem Solving stochastic min-max problems with strong convexity and strong concavity.
method Epoch-wise stochastic gradient descent ascent method (Epoch-GDA) without additional assumptions.
result Achieves the optimal rate of for the duality gap of general SCSC min-max problems.
The paper bounds the min-max width of embedded circles on spheres and manifolds.
problem Bounding the min-max width of embedded circles on spheres and manifolds.
method Inducing a sweepout by pairs of points in embedded circles from a given sweepout of the sphere by closed curves.
result Lower bounds for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles.
Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
Bayesian optimization methods improved for min max optimization problems.
problem Min-max optimization for unknown functions.
method Extended Bayesian optimization to min-max problems with new acquisition functions.
result Improved acquisition functions lead to better solutions.
In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension . We characterize the …
The study proves a generic multiplicity one theorem for -invariant minimal hypersurfaces.
problem Proving a generic multiplicity one theorem for -invariant minimal hypersurfaces.
method Equivariant min-max theory and analysis of -homology classes.
result Shows a generic multiplicity one theorem for -invariant minimal hypersurfaces.
Constructs cmc doublings of minimal surfaces via min-max theory.
problem Construct cmc doublings of minimal surfaces.
method Uses min-max theory and catenoid estimate.
result Constructs ε-cmc doublings of Σ for small ε > 0.
This research proves that two min-max theories for hypersurfaces are equivalent.
problem Comparing two min-max theories for hypersurfaces.
method Developed and proved the equivalence of Almgren-Pitts and Allen-Cahn min-max theories.
result The Almgren-Pitts widths and Allen-Cahn widths are equivalent.
This study reveals a Min-Max property in LeNet's convolutional layers, enhancing adversarial robustness.
problem Uncertainty in the connection weights of convolutional layers in neural networks.
method Demonstrates the Min-Max property through back propagation-based training and a simplified convolution formulation.
result The Min-Max property improves adversarial robustness, indicating a stronger uncertainty in the model parameters.
New Gaussian min-max theorem extends classical results to non-i.i.d. Gaussian matrices.
problem Extending classical Gaussian min-max theorems to non-i.i.d. Gaussian matrices.
method Identifying a new pair of Gaussian processes that satisfy comparison inequalities.
result New Gaussian min-max and convex Gaussian min-max theorems with applications in multi-source Gaussian regression and binary classification.
New algorithm solves non-convex, non-differentiable min-max games.
problem Limited theoretical understanding of non-smooth min-max games.
method Proximal gradient descent-ascent algorithm for convex-strongly convex games.
result Algorithm converges to ε-Nash equilibrium with polynomial gradient evaluations.
Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
problem Finding generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
method Weyl asymptotic law for -equivariant volume spectrum, generic density result.
result Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
We characterize the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coin…
New geometric invariant from min-max width of spheres on Riemannian 2-spheres.
problem Understanding the min-max width of spheres associated to distance functions.
method Application of min-max methods to pairs of points on Riemannian 2-spheres.
result The min-max width does not always equal half the length of a simple closed geodesic.
Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to ac…
Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.
problem Constructing stable anisotropic minimal surfaces in 3-manifolds.
method Anisotropic min-max theory, removable singularity theorems.
result Constructs stable anisotropic minimal surfaces in 3-manifolds without singularities.
In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature . Moreover…
The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.
problem Finding smooth anisotropic minimal surfaces in closed 3-manifolds.
method Min-max construction with elliptic integrands, uniform upper bound for density ratios.
result Obtains a smooth anisotropic minimal surface in a closed 3-manifold.
In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in \cite{A2}\cite{P} corresponding to the fundamental class of a Riemannian manifold of positive Ricci curvature with . We characterize the Morse index, area and multiplicity of this min-max hyp…