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.
The study uncovers the breakdown of Gaussian universality in high-dimensional empirical risk minimization.
problem Understanding the breakdown of Gaussian universality in high-dimensional empirical risk minimization.
method Extending the Convex Gaussian Min-Max Theorem to non-Gaussian settings, deriving asymptotic min-max characterizations, and proving asymptotic equivalence of regularizers.
result The projection of the ERM estimator onto a test covariate approximately follows a Gaussian convolution under certain conditions.
We analyze learning curves of RF models with convex regularization and derive precise asymptotic expressions.
problem Understanding the learning curves of RF models with general convex regularization.
method Novel multi-level application of the convex Gaussian min max theorem (CGMT) to compute precise asymptotic expressions.
result Precise asymptotic expressions for learning curves of RF models with separable strongly convex regularization or ℓ1 regularization. We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut theorem for boundary regions, applied recently to develop a "bit-thread" interpretation of holographic entanglement entropies. We also prove various properties of the max flow…
Deep random feature models are analyzed for their performance with exact asymptotic expressions.
problem Understanding the performance of deep random feature models.
method Established a novel universality result and used the convex Gaussian Min-Max theorem.
result Exact asymptotic expressions for the performance of deep random feature models are derived.
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.
The study proves the existence of free boundary minimal disks in convex regions.
problem Proving the existence of free boundary minimal disks in convex regions.
method Based on a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.
result Existence of at least three embedded free boundary minimal disks in strictly convex domains with nonnegative Ricci curvature.
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 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. Max flow/min cut theorem extended to currents and topology.
problem Continuous max flow/min cut theorem for complex domains.
method Continuous analogue of max flow/min cut theorem considering topology.
result Continuous max flow/min cut theorem proven for currents and laminations.
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.
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.
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 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…
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.
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.
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.
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.
Study max- and min-stability under first-order stochastic dominance, finding new functional characterizations.
problem Understanding max- and min-stability in stochastic dominance.
method Representation theorem for functionals satisfying max-stability, combining max- and min-stability to define Lambda-quantiles.
result New characterizations of functionals, including Lambda-quantiles, in finance and political science.
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. The paper explains how simple methods can converge to optimal solutions in complex neural games.
problem Finding optimal solutions in neural games with non-convex objectives.
method Theoretical framework using hidden convexity and overparameterization, with path-length bounds and PŁ conditions.
result Simple gradient methods can converge to Nash equilibria in non-convex min-max games under certain conditions.
Proposes active sampling for improving fairness in machine learning.
problem Improving fairness in machine learning models, especially for disadvantaged groups.
method Simple active sampling and reweighting strategies for min-max fairness.
result Proves the rate of convergence to a min-max fair solution for convex problems.
New saddle network architectures preserve convex-concave geometry in optimization problems.
problem Optimization models with convex x and concave y components.
method Structured separable decomposition and saddle network architectures.
result Proven one-dimensional approximation theorem and high accuracy on various test functions.
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 (Mn+1,g) of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension 7. We characterize the …
Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.
problem Existence of Yamabe metrics on conical 4-manifolds with singular points.
method Min-max scheme adapted to singular setting, leveraging recent positive mass theorems.
result Existence of Yamabe metrics on conical 4-manifolds with finitely-many singular points.
Riemannian algorithms converge at Euclidean rates for geodesically convex-concave problems.
problem Min-max optimization on Riemannian manifolds.
method RCEG method and RGDA for geodesically strongly-convex-concave problems.
result RCEG achieves linear convergence rate in geodesically strongly-convex-concave cases.
The study bounds the topology of free boundary minimal surfaces in 3D manifolds.
problem Understanding the topology of free boundary minimal surfaces in compact 3D manifolds.
method Establishing general bounds on the topology via min-max methods and analyzing varifolds.
result The first Betti number is lower semicontinuous in the limit of min-max sequences.
Let (Mn+1,∂M,g) be a compact manifold with non-negative Ricci curvature, convex boundary and 2≤n≤6. We show that the min-max minimal hypersurface with respect to one-parameter families of hypersurfaces in (M,∂M) is orientable, of index one and multiplicity one.
Paper tackles fast convergence for non-convex strongly-concave min-max problems.
problem Non-convex strongly-concave min-max problems in deep learning.
method Proximal stage-based method with PL condition for faster convergence.
result Established fast convergence in primal objective gap and duality gap.
Proves existence of minimal surfaces with fixed boundary contact angle.
problem Existence of minimal surfaces with fixed boundary contact angle.
method Min-max construction in the spirit of Almgren-Pitts for the capillarity functional.
result Existence of minimal surfaces in a bounded convex subset of R^3 with fixed boundary contact angle.
In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in mach…
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.
While classic work in convex-concave min-max optimization relies on average-iterate convergence results, the emergence of nonconvex applications such as training Generative Adversarial Networks has led to renewed interest in last-iterate convergence guarantees. Proving last-iterate convergence is challenging because ma…
The abstract finds conditions for creating curves of constant curvature.
problem Finding conditions for closed embedded curves of constant curvature.
method Using Almgren-Pitts min-max method for geodesic curvature.
result Closed embedded curves of any prescribed constant curvature found in S2. 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. PURE-CD algorithm proves complexity bounds for convex-concave problems.
problem Solving convex-concave min-max problems with bilinear coupling.
method Primal-dual algorithm with random extrapolation and coordinate descent (PURE-CD).
result Complexity bounds match or improve existing results for dense and sparse problems.
New study shows min-max algorithms can converge to non-stationary points.
problem Challenges in min-max optimization due to periodic cycles and spurious attractors.
method Analyzed state-of-the-art algorithms and heuristics in non-convex/non-concave problems.
result Spurious attractors can prevent min-max algorithms from reaching true optima.
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.
Entropy is a natural geometric quantity measuring the complexity of a surface embedded in R3. 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 …
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
Theory for capillary surfaces in 3-manifolds with smooth boundary.
problem Existence and multiplicity of capillary surfaces with given mean curvature and contact angle.
method Min-max theory applied to capillary surfaces in 3-manifolds.
result Existence of nontrivial, smooth, almost properly embedded surfaces with constant mean curvature and contact angle.
Study finds geodesic networks for surfaces with convex boundary.
problem Finding geodesic networks for surfaces with convex boundary.
method Investigates free boundary geodesic networks in surfaces with non-negative sectional curvature and convex boundary.
result Existence of a geodesic network realizing the first width of a surface with non-negative sectional curvature and strictly convex boundary.
Paper tackles multi-block min-max optimization with applications in deep AUC maximization.
problem Multi-block min-max bilevel optimization with non-convex strongly-concave upper level and strongly convex lower level.
method Single-loop randomized stochastic algorithm for constant number of blocks per iteration.
result Sample complexity of O(1/ε^4) for finding ε-stationary point, matching optimal complexity.
Let M be a stratum of a compact stratified space A. It is equipped with a general adapted metric g, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, g has a general type, which is an extension of the type of an adapted metric. A restriction on this …
In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus g≥2. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a …
Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.
problem Proving multiplicity one for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
method Developed existence and regularity theory for free boundary hypersurfaces with prescribed mean curvature, including Morse index bounds.
result Proved multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
The study finds at least 2 free-boundary minimal disks in convex 3-balls.
problem Finding minimal disks in convex 3-balls.
method Combining mean curvature flow, min-max theory, and degree theory.
result Existence of at least 2 free-boundary minimal disks in convex 3-balls for generic metrics.
Previously, Cristofaro-Gardiner, Hutchings and Ramos have proved that embedded contact homology (ECH) capacities can recover the volume of a contact 3-manifod in their paper "the asymptotics of ECH capacities" . There were two main steps to proving this theorem: The first step used an estimate for the energy of min-max…