The paper compares eigenvalues in curved spaces with spherical models.
problem Comparing eigenvalues in curved spaces with spherical models.
method Min-max formulas and pointwise curvature comparisons.
result Eigenvalue comparisons in curved spaces generalize known results.
Paper defines a new functional for spinors on Euclidean manifolds.
problem No specific problem stated; focuses on a new functional.
method Variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.
result Ricci flow is the gradient flow of the new functional.
Study properties of Black-Scholes equation solutions for puttable bonds with credit risk.
problem Properties of solutions to Black-Scholes equation for puttable bonds with credit risk.
method Solution representation, min-max estimation, gradient estimates, strict monotonicity analysis.
result Derivation of analytical pricing formulae for puttable bonds with credit risk.
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.
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.
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 2π2 and 8π? 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 2π2 and 8π. Proves multiplicity one for min-max minimal hypersurfaces in specific manifolds.
problem Proving multiplicity one for min-max minimal hypersurfaces in specific manifolds.
method Using min-max theory for hypersurfaces with prescribed mean curvature and approximating min-max values.
result Confirms a conjecture by Marques-Neves for min-max minimal hypersurfaces in bumpy metrics.
Characterizes Zoll metrics via min-max values.
problem Characterizing Zoll Riemannian metrics.
method Uses min-max values in a loop space.
result Two min-max values coincide for Zoll metrics.
Paper develops min-max theory for CMC hypersurfaces.
problem Constructing constant mean curvature hypersurfaces.
method Min-max theory applied to arbitrary closed manifolds.
result Existence of nontrivial, smooth, closed, almost embedded CMC hypersurfaces.
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.
Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.
problem Existence and finiteness of G-invariant minimal hypersurfaces. method Equivariant min-max theory, compactness theorem, bumpy metrics theorem.
result Generalization of Morse index estimates to equivariant setting.
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.
The study bounds Morse indices of Willmore spheres in relation to min-max sweep-outs.
problem Estimating Morse indices of Willmore spheres.
method Analyzing the sum of Morse indices of Willmore spheres in min-max sweep-outs.
result At most one Willmore sphere can have index 1 among those realising min-max sphere eversion.
We rigorously prove statistical physics predictions for non-convex GLMs in high dimensions.
problem Analyzing high-dimensional optimization problems in non-convex Generalized Linear Models.
method Developed a systematic framework using the Gaussian Min-Max Theorem and AMP to rigorously prove replica-symmetric formulas.
result Validated statistical physics predictions for non-convex GLMs, aligning with physicist's conjectures.
New Morse index bounds for min-max minimal hypersurfaces solved multiplicity problem.
problem Finding Morse index bounds for min-max minimal hypersurfaces.
method Advanced Min-max Theory for the area functional, including new Morse index bounds.
result First general Morse index bounds for min-max minimal hypersurfaces.
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 RP3 and lens spaces. Study min-max hypersurface in positive Ricci curvature manifolds.
problem Characterize the properties of min-max hypersurface in positive Ricci curvature manifolds.
method Apply Almgren-Pitts-Schoen-Simon min-max theory, discretization theorem, and Morse theory.
result The min-max hypersurface is either orientable with Morse index one or a double cover of a non-orientable stable minimal hypersurface.
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 3≤n+1≤7. 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.
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.
Find geodesics with free boundary on submanifolds.
problem Finding geodesics with free boundary on submanifolds of Riemannian manifolds.
method Developed a modified Birkhoff curve shortening process and min-max theory for free boundary variational problems.
result Strong min-max approximation result for free boundary geodesics.
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 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 O(1/T) 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.
Develops equivariant min-max theory for minimal surfaces in S^3.
problem Finding minimal surfaces in S^3 with specific symmetries.
method Equivariant min-max theory as proposed by Pitts-Rubinstein.
result Produces new and known minimal surfaces in S^3.
Generic min-max theory proves existence of hypersurfaces with specific mean curvature.
problem Proving existence of hypersurfaces with prescribed mean curvature for generic functions.
method Generic min-max theory applied to smooth prescription functions.
result Existence of nontrivial, smooth, closed hypersurfaces with specific mean curvature.
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.
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.
The study proves a generic multiplicity one theorem for G-invariant minimal hypersurfaces.
problem Proving a generic multiplicity one theorem for G-invariant minimal hypersurfaces. method Equivariant min-max theory and analysis of G-homology classes. result Shows a generic multiplicity one theorem for G-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 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 insights into gradient descent and ascent dynamics in min-max optimization.
problem Understanding the convergence and limit points of gradient descent and ascent methods in min-max optimization problems.
method Characterization of limit points using dynamical systems perspective for GDA and OGDA.
result Both GDA and OGDA dynamics avoid unstable critical points and have a superset of local min-max solutions.
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.
Proves min-max theory for constant geodesic curvature curves on closed surfaces.
problem Prescribing mean curvature on surfaces with constant geodesic curvature.
method Min-max theory applied to classify blowups and ensure almost embedded solutions.
result Produces a solution with constant geodesic curvature c on closed surfaces. 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 G-equivariant volume spectrum, generic density result. result Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
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.
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.
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 (Mn+1,g) of positive Ricci curvature with 2≤n≤6. We characterize the Morse index, area and multiplicity of this min-max hyp…
Unified framework for adversarial attacks using min-max optimization.
problem Improving adversarial robustness and attack generation.
method General framework of min-max optimization over multiple domains.
result Substantial attack improvement and robustness improvement.
Improved algorithms for convex-concave min-max optimization and monotone variational inequalities.
problem Efficiently solving constrained convex-concave min-max problems and monotone variational inequalities.
method Higher-order methods achieving iteration complexities of O(1/T^{ rac{p+1}{2}}) for p-th order derivatives.
result Achieved improved convergence rates for min-max and monotone variational inequalities.
New algorithm solves min-max optimization problems in a decentralized manner.
problem Solving min-max saddle point games in a decentralized and adaptive manner.
method Developed a decentralized adaptive momentum (DADAM3) algorithm for min-max optimization. result DADAM3 achieves non-asymptotic rates of convergence for finding Nash equilibrium points. Optimizes solving complex min-max problems with stochastic and nonconvex elements.
problem Min-max problems with stochastic and nonconvex elements.
method Combines conic nonexpansiveness, refined inexact Halpern iteration, and multilevel Monte Carlo estimator.
result Optimal or best-known complexity guarantees for $ρ< rac{1}{L}$, improving previous results.