arXiv research

Paper improves Morse index bound for hypersurfaces.

Study shows instability of specific cone solutions in high-dimensional spaces.

We consider the problem of two-player zero-sum games. This problem is formulated as a min-max Markov game in the literature. The solution of this game, which is the min-max payoff, starting from a given state is called the min-max value of the state. In this work, we compute the solution of the two-player zero-sum game…

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…

The min-max kernel is a generalization of the popular resemblance kernel (which is designed for binary data). In this paper, we demonstrate, through an extensive classification study using kernel machines, that the min-max kernel often provides an effective measure of similarity for nonnegative data. As the min-max ker…

We use min-max techniques to produce nontrivial solutions $u_ε:M\to \mathbb{R}^2$ of the Ginzburg-Landau equation $Δu_ε+\frac{1}{ε^2}(1-|u_ε|^2)u_ε=0$ on a given compact Riemannian manifold, whose energy grows like $|\logε|$ as $ε\to 0$. When the degree one cohomology $H^1_{dR}(M)=0$, we show that the energy of these s…

We employ min-max techniques to show that the unit ball in $\mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.

Existence proof of noncompact self-shrinkers with arbitrary genus.

A novel feature selection method for SVM improves model accuracy and interpretability.

The paper finds closed hypersurfaces with prescribed mean curvature in non-compact manifolds.

Study shows strong min-max principle for phase transitions.

Equity-Transformer solves NP-hard min-max routing problems efficiently.

Upper bound for Morse index of min-max varifolds.

Localized min-max method proves minimal hypersurface existence.

The paper solves min-max widths on a 3-sphere and strengthens multiplicity theorems.

Study introduces statistical mechanics for min-max problems.

Adaptive momentum method solves non-convex min-max problems.

Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.

New proof of Smale conjecture for RP^3 and lens spaces using min-max theory.

We study global variational properties of the space of solutions to $-\varepsilon^2Δu + W'(u)=0$ on any closed Riemannian manifold $M$. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show t…

New methods solve min-max problems on manifolds using Riemannian Hamiltonians.

Bound on equivariant index for min-max surfaces.

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…

Survey of advances in non-convex min-max optimization for applications.

We reformulate LIPs as min-max problems for easier solution.

The paper bounds the min-max width of embedded circles on spheres and manifolds.

Study confirms a 2-sphere metric with three geodesics of minimal length.

Bayesian optimization methods improved for min max optimization problems.

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 $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the …

The study proves a generic multiplicity one theorem for $G$-invariant minimal hypersurfaces.

Constructs cmc doublings of minimal surfaces via min-max theory.

This research proves that two min-max theories for hypersurfaces are equivalent.

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold ($2\le n\le 6$) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max methods: they have index at most $1$. We apply this to obtain a lower area bound for su…

This study reveals a Min-Max property in LeNet's convolutional layers, enhancing adversarial robustness.

New Gaussian min-max theorem extends classical results to non-i.i.d. Gaussian matrices.

New algorithm solves non-convex, non-differentiable min-max games.

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.

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.

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 $c$. Moreover…

The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.

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 $(M^{n+1}, g)$ of positive Ricci curvature with $2\leq n\leq 6$. We characterize the Morse index, area and multiplicity of this min-max hyp…

We show that the sum of the Morse indices of the Willmore spheres realising the width of Willmore type sweep-outs is bounded by the number of the parameters of the min-max. As an application, we deduce that among the true Willmore spheres realising the min-max sphere eversion, at most one of them one has index 1, while…

Epoch gradient descent method (a.k.a. Epoch-GD) proposed by Hazan and Kale (2011) was deemed a breakthrough for stochastic strongly convex minimization, which achieves the optimal convergence rate of $O(1/T)$ with $T$ iterative updates for the {\it objective gap}. However, its extension to solving stochastic min-max pr…

Improved algorithms for convex-concave min-max optimization and monotone variational inequalities.

New algorithm solves min-max optimization problems in a decentralized manner.