Algorithm finds minimal standardizer of parabolic subgroups in Artin-Tits groups.
problem Computing minimal standardizer of parabolic subgroups in Artin-Tits groups.
method Geometrical and algebraic algorithms to compute pn-normal form of a central element. result Minimal standardizer computation simplified to pn-normal form. Study minimal networks on spheres and balls near standard metrics.
problem Existence of minimal networks in spheres and balls with metrics close to standard.
method Finite-dimensional reduction method, inspired by configuration of networks and triods.
result Existence of minimal networks in spheres and balls for metrics close to standard.
Simplifies risk minimization combining mean and standard deviation.
problem Minimizing mean and standard deviation under heavy-tailed losses.
method Adapting robust mean estimation technique to include standard deviation.
result Simple approach performs as well or better than alternative risk criteria.
In 1993, Y.-G. Oh proposed a problem whether standard Lagrangian tori in C^n are volume minimizing under Hamiltonian isotopies of C^n. In this article, we prove that most of them do not have such property if the dimension n is greater than two. We also discuss the existence of Hamiltonian non-volume minimizing Lagrangi…
Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.
problem Optimizing total σ2-curvature on spheres with positive scalar curvature. method Analyzes metrics conformal to the standard sphere, uses Sobolev norms to measure closeness.
result Near-minimizers of total σ2-curvature are almost the standard metric (up to Möbius transformations). This work proposes SDI regularization to improve adversarial robustness.
problem Improving adversarial robustness of deep neural networks.
method SDI regularization term to complement adversarial training.
result Combining SDI with AT variants enhances robustness and generalization.
We describe a 3-parametric family K of properly embedded minimal tori with four parallel ends in quotients of R3 by two independent translations, which we will call the \textit{Standard Examples.} These surfaces generalize the examples given by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}.…
It is shown that parts of planes, helicoids and hyperbolic paraboloids are the only minimal surfaces ruled by geodesics in the three dimensional Riemannian Heisenberg group. It is also shown that they are the only surfaces in the three dimensional Heisenberg group whose mean curvature is zero with respect to both of th…
Standard embeddings of Schubert varieties in specific manifolds characterized.
problem Characterizing standard embeddings of Schubert varieties in rational homogeneous manifolds.
method Characterization through varieties of minimal rational tangents.
result Characterization of standard embeddings of smooth Schubert varieties.
The study finds counterexamples to curvature estimates for minimizing surfaces.
problem Curvature estimates for minimizing surfaces in metric convergence.
method Constructing sequences of smooth minimizing surfaces in metrics converging to Euclidean.
result Found counterexamples with diverging L2 norm of second fundamental form. Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.
problem Creating space-filling shapes without sharp corners.
method Edge bending algorithm to deform polyhedral tilings into soft tilings.
result Soft tilings derived from minimal surfaces can be continuously transformed into one another.
Standard cohomology of Courant algebroids identified via minimal models.
problem Cohomology of Courant algebroids.
method Minimal model construction and Hodge-to-de Rham spectral sequence.
result Standard cohomology of Courant algebroids identified with function space cohomology.
Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination w…
We present conditions on the Ricci curvature for complete, oriented, minimal submanifolds of Euclidean space, as well as the standard unit sphere, when the Gauss maps are bounded embeddings.
Study finds minimal coloring numbers for torus links using rack colorings.
problem Determining the minimal number of colors for torus link diagrams.
method Using rack colorings on link diagrams to classify minimal colorings.
result Complete classifications of Z-colorings by four colors. Transforms data for interpretable recommender systems.
problem Building an interpretable recommender system for personalized content and promotions.
method Loss-preserving transformation to standard interpretable multi-class classification algorithms.
result Minimizing standard misclassification penalty in the new space is equivalent to minimizing custom cost function.
New method outperforms standard procedures in heavy-tailed problems.
problem Regression function estimation under heavy-tailed conditions.
method Regularized risk minimization procedure based on median-of-means tournaments.
result The new procedure achieves near optimal accuracy and confidence in heavy-tailed problems.
Efficiently minimizes regret in non-convex games with gradient-based methods.
problem Computational intractability of standard regret minimization in non-convex games.
method Defining a new notion of regret and using gradient-based optimization methods.
result Achieves optimal regret, leading to convergence to equilibrium.
In this note, we investigate the well-known Yau rigidity theorem for minimal submanifolds in spheres. Using the parameter method of Yau and the DDVV inequality verified by Lu, Ge and Tang, we prove that if M is an n-dimensional oriented compact minimal submanifold in the unit sphere Sn+p(1), and if $K_{M}\geq\…
New minimal link diagrams found, including torus links and homogeneous ones.
problem Finding minimal link diagrams with new classes.
method Morton-Franks-Williams inequality approach.
result New classes of minimal link diagrams, including previously unproven ones.
Freedman, He, and Wang, conjectured in 1994 that the Mobius energy should be minimized, among the class of all nontrivial links in Euclidean space, by the stereographic projection of the standard Hopf link. We prove this conjecture using the min-max theory of minimal surfaces.
Most Lagrangian torus orbits in complex projective space are not volume minimizing.
problem Identifying volume minimizing Lagrangian orbits in complex projective spaces.
method Analyzing Hamiltonian isotopies and volume minimization properties of Lagrangian torus orbits.
result Almost all Lagrangian torus orbits are not Hamiltonian volume minimizing.
New algorithm SELECT minimizes satisficing regret in bandits.
problem Minimizing regret in bandit optimization with satisficing arms.
method SELECT algorithm for satisficing regret minimization.
result SELECT achieves constant expected satisficing regret.
We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the 3-torus is the standard double bubble from R^3.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
We show that the standard minimal genus Heegaard splitting of (closed orientable surface)\times S^1 is a critical Heegaard splitting.
New proof of a unique 3-part partition in 8D space.
problem Existence of a non-standard isoperimetric partition in high dimensions.
method Analytical proof showing existence of a specific partition.
result Existence of a non-standard isoperimetric partition in 8D space.
Hexagonal tilings minimize perimeter with unequal volumes.
problem Finding optimal tessellations with unequal cell volumes.
method Minimizing perimeter functionals for different classes of problems.
result Hexagonal tilings are optimal among partitions with almost equal areas.
We consider the minimal entropy problem, namely the question of whether there exists a smooth metric of minimal entropy, for certain classes of 3-manifolds. Among other resulsts, we show that if M is a closed, orientable, geometrizable 3-manifold with zero simplicial volume, then the minimal entropy can be solved for M…
Proves Miyaoka-Yau inequality for minimal pairs using Kähler-Einstein metrics.
problem Proving Miyaoka-Yau inequality for minimal pairs.
method Using Kähler-Einstein metrics with conical and cuspidal singularities.
result Proves Miyaoka-Yau inequality for all minimal pairs.
A new method for 1-bit matrix completion that is faster and more accurate.
problem Estimating a low-rank matrix from binary observations.
method Majorization-Minimization Gauss-Newton (MMGN) method.
result MMGN outperforms existing methods in accuracy and speed.
Solves empirical risk minimization for relational data using graph sampling.
problem Empirical risk minimization for relational data.
method Graph sampling theory, stochastic gradient descent, automatic differentiation.
result Automatic unbiased stochastic gradients for relational data.
We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau prob…
Let M be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric ≥0. We suppose that M is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let Σ be a compact connected and orientable surface immersed in M which is a stable constan…
In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of crit…
A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einst…
The paper constructs stable minimal hypersurfaces with specific singularities.
problem Creating minimal hypersurfaces with controlled singularities.
method Constructing hypersurfaces with a given singular set in a modified Euclidean space.
result Embedded minimal hypersurfaces with stable properties and specified singularities.
This study connects Jacobian regularization to adversarial robustness and improves generalization.
problem Adversarial attacks make deep neural networks vulnerable.
method Developed a connection between Jacobian regularization and adversarial training, and established robust generalization gaps.
result Jacobian norms are related to both standard and robust generalization.
JTT improves model worst-group accuracy without group annotations.
problem Low worst-group accuracy in standard ERM models with spurious correlations.
method Two-stage approach: first ERM, then upweight misclassified examples.
result JTT closes 75% of the gap in worst-group accuracy compared to group DRO.
We study the problem of online learning with a notion of regret defined with respect to a set of strategies. We develop tools for analyzing the minimax rates and for deriving regret-minimization algorithms in this scenario. While the standard methods for minimizing the usual notion of regret fail, through our analysis …
Let Dn denote the n-punctured disk in the complex plane, where the punctures are on the real axis. An n-braid α is said to be \emph{reducible} if there exists an essential curve system $\C$ in Dn, called a \emph{reduction system} of α, such that $α*\C=\C$ where $α*\C$ denotes the action of the braid α o…
The study of stable and index compact minimal submanifolds in Berger spheres.
problem Stability and index of compact minimal submanifolds in Berger spheres.
method Analyzing stability and index properties of compact minimal submanifolds in Berger spheres.
result Stable compact minimal submanifolds exist in Berger spheres for specific values of τ, and their classification is provided.
Unified algorithm for minimizing composite functions with flexible design.
problem Minimizing composite functions with specific structural properties.
method Unified accelerated algorithm for complementary composite minimization.
result Near-optimal algorithms for various optimization problems.
IRM fails to improve over standard methods in complex settings.
problem Learning invariant features for out-of-distribution generalization.
method Analysis of Invariant Risk Minimization (IRM) and related approaches under a general model.
result IRM can fail catastrophically in non-linear settings, even when test data are similar to training distribution.
Efficiently constructs prediction bands with minimal assumptions.
problem Uncertainty quantification for nonparametric, heteroscedastic data.
method Semi-definite programming for data-adaptive prediction bands.
result Strong non-asymptotic coverage properties with minimal distributional assumptions.
This paper shows how to learn variational inequalities fast with strong monotonicity.
problem Learning variational inequalities efficiently.
method Extending convex optimization techniques to variational inequalities with strong monotonicity.
result Fast generalization rates of Θ(1/ε) for learning variational inequalities. Extends fair empirical risk minimization to continuous sensitive attributes.
problem Avoiding unfair influence of sensitive information in regression.
method Generalized fair empirical risk minimization framework for continuous sensitive attributes.
result Derives learning guarantees for statistical consistency in risk and fairness.
New algorithm minimizes worst-case regret in uncertain, time-varying dynamics.
problem Model-based policy learning in uncertain, time-varying dynamics.
method Planning regret metric and iterative algorithm for minimizing it.
result Empirical evidence shows the proposed algorithm outperforms existing methods.