arXiv research

Extends Minkowski stability proof to minimal decay assumptions.

New method creates vacuum data at minimal and borderline decay thresholds.

Extends global stability of Minkowski spacetime to minimal decay assumptions.

Study optimizes decay estimates for minimizing currents in submanifolds.

SGD and weight decay encourage neural networks to learn low-rank weight matrices.

New inequality shows energy growth and decay in geometric problems.

We study minimal graphic functions on complete Riemannian manifolds $\Si$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with …

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…

Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.

We prove a general asymptotic decay lemma which is applicable in various contexts. As an example, the general theorem is shown to give lower growth estimates for entire and exterior solutions of the minimal surface equation.

Study on scalar curvature decay on non-compact manifolds linked at infinity.

Study area minimizing currents in Riemannian manifolds, proving unique structure and decay.

Study finds solutions to inequality decay to zero on warped cylinders.

New method improves smoothness of minimizing currents near singular points.

New height estimate for area minimizing currents, leading to unique tangent cones and decay properties.

We study the asymptotic Dirichlet problem for $f$-minimal graphs in Cartan-Hadamard manifolds $M$. $f$-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of $f$-minimal graphs with pre…

We present an explicit example of a fast decaying solution to the modified Novikov--Veselov equation with a one-point singularity in the space-time. It is constructed by using the geometrical interpretation of the Moutard transformation of solutions to this equation and the Enneper minimal surface.

Forward construction of vacuum initial data with limited decay

Kernel Density Estimation is a very popular technique of approximating a density function from samples. The accuracy is generally well-understood and depends, roughly speaking, on the kernel decay and local smoothness of the true density. However concrete statements in the literature are often invoked in very specific …

Study on optimal ReLU networks with weight decay for interpolation.

Study on singularities in area-minimizing currents, proving unique tangent cones and rectifiability.

SHIFT framework identifies subgroups with large ML model performance decay.

In this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This removable singularity theorem is the key result used in our proof that a complete, embedded minimal surface in $\mathbb{R}^3$ with quadratic decay of curvature ha…

Paper tackles dynamic pricing in a geometrically decaying environment, achieving better occupancy with lower rates.

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…

Adaptive LR improves neural network Lipschitz regularity without slowing convergence.

Novel Adam-family method with decoupled weight decay for training neural networks.

Model proposes neural network for continuous time dynamics with inductive biases.

Stochastic (sub)gradient methods require step size schedule tuning to perform well in practice. Classical tuning strategies decay the step size polynomially and lead to optimal sublinear rates on (strongly) convex problems. An alternative schedule, popular in nonconvex optimization, is called \emph{geometric step decay…

We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup $H$. We further assume that $H$ is not consisting only of lifts with respect to any one covering. Then w…

WSD schedule improves model training efficiency by adapting learning rates dynamically.

We show that a complete Euclidean submanifold with minimal index of relative nullity $ν_0>0$ and Ricci curvature with a certain controlled decay must be a $ν_0$-cylinder. This is an extension of the classical Hartman cylindricity theorem.

Survey on stability of Minkowski spacetime in relativity.

This paper concerns some stability properties of higher dimensional catenoids in $\rr^{n+1}$ with $n\ge 3$. We prove that higher dimensional catenoids have index one. We use $δ$-stablity for minimal hypersurfaces and show that the catenoid is $\frac 2n$-stable and a complete $\frac 2n$-stable minimal hypersurface is a …

In a 2004 paper, Lindblad demonstrated that the minimal surface equation on $\mathbb{R}l^{1,1}$ describing graphical time-like minimal surfaces embedded in $\mathbb{R}^{1,2}$ enjoy small data global existence for compactly supported initial data, using Christodoulou's conformal method. Here we give a different, geometr…

The paper connects neural collapse and low-rank bias in networks with L2 regularization.

We introduce a new weight-decay scaling rule to maintain sublayer gains across different widths in modern scale-invariant architectures.

SGD's performance improves with critical batch size, minimizing SFO complexity.

Neural networks minimize error with shallow ReLU models for function estimation.

We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has $L^2$-bounded second fundamental form and satisfies a weak power growth on the area. We…

We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields …

Paper questions RNN and LSTM's long-term memory and introduces a new definition.

A novel k-means method for MNAR data improves clustering accuracy.

New method for training deep neural networks with regularization, converging to better generalization.

We study dynamic hedging of counterparty risk for a portfolio of credit derivatives. Our empirically driven credit model consists of interacting default intensities which ramp up and then decay after the occurrence of credit events. Using the Galtchouk-Kunita-Watanabe decomposition of the counterparty risk price paymen…

Characterizes eigenfunctions of Lawson-Osserman cone and proves its integrability.

In this paper we prove two theorems. The first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature (possibly zero) in a homogeneously regular Riemannian three-manifold, in any small neighborhood of a point of large almost-maximal curvature. We next apply t…

Unique solutions found for wave-like decaying null infinity equations.