arXiv research

Paper proves Gromov's conjecture on manifolds with certain group properties.

Constructs a support-preserving homotopy for differential forms with boundary decay estimates.

We give conditions which imply that a complete noncompact manifold with quadratic curvature decay has finite topological type. In particular, we find links between the topology of a manifold with quadractic curvature decay and some properties of the asymptotic cones of such a manifold.

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

We study a notion of a Lipschitz, permutation-invariant "centroid" for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).

The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.

Paper shows existence of vortex solutions with specific decay properties.

Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery …

Improved online prediction with guaranteed coverage.

We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such s…

Regularizers change the geometric properties of loss functions in neural networks.

Step decay schedules improve convergence in non-convex optimization.

A new accelerated method with simpler momentum update rules.

We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10…

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

This paper constructs a class of complete Kähler metrics of positive holomorphic sectional curvature on ${\bf C}^n$ and finds that the constructed metrics satisfy the following properties: As the geodesic distance $ρ\to\infty,$ the volume of geodesic balls grows like $O(ρ^{\frac{2(β+1)n}{β+2}})$ and the Riemannian scal…

This paper is motivated by the non-linear stability problem for the expanding region of Kerr de Sitter cosmologies in the context of Einstein's equations with positive cosmological constant. We show that under dynamically realistic assumptions the conformal Weyl curvature of the spacetime decays towards future null inf…

New method finds precise late-time behavior of wave equations.

The paper constructs Ricci-flat Kähler manifolds with specific decay properties.

Constructs expanding gradient Ricci solitons with unique properties.

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundam…

Study proves global existence and decay for complex wave equations.

The study explores dilating set properties across Euclidean and hyperbolic geometries.

The paper analyzes Teukolsky equations on Kerr backgrounds, proving boundedness and decay of solutions.

New method for estimating mean in SS inference with selection bias and decaying overlap.

This paper analyzes convergence of large-scale Transformers with weight decay.

Deep ReLU networks approximate as well as shallow ones in kernel regimes.

In this paper we study the behaviour of the continuous spectrum of the Laplacian on a complete Riemannian manifold of bounded curvature under perturbations of the metric. The perturbations that we consider are such that its covariant derivatives up to some order decay with some rate in the geodesic distance from a fixe…

The salient properties of large empirical covariance and correlation matrices are studied for three datasets of size 54, 55 and 330. The covariance is defined as a simple cross product of the returns, with weights that decay logarithmically slowly. The key general properties of the covariance matrices are the following…

New method constructs flat initial data for Einstein's equations.

Study of flows on circle bundles over translation surfaces, showing decay of correlations.

The paper studies Teichmüller TQFT for hyperbolic knots, proving exponential decay of partition functions.

New theory sharpens Q-learning with LDTZ rate, proving it's best of both worlds.

This paper investigates the effectiveness of decoupled weight decay at the start of training.

The spectral $k$-support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank $k$ matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral $(k,p)$-support norm, whose additional para…

Nuclear magnetic resonance (NMR) spectroscopy exploits the magnetic properties of atomic nuclei to discover the structure, reaction state and chemical environment of molecules. We propose a probabilistic generative model and inference procedures for NMR spectroscopy. Specifically, we use a weighted sum of trigonometric…

We review our recent work on linear stability for scalar perturbations of Kerr spacetimes, that is to say, boundedness and decay properties for solutions of the scalar wave equation \Box_gψ = 0 on Kerr exterior backgrounds. We begin with the very slowly rotating case |a| \ll M, where first boundedness and then decay ha…

Improves model classification accuracy in black-box settings.

We analyze dropout in deep networks with rectified linear units and the quadratic loss. Our results expose surprising differences between the behavior of dropout and more traditional regularizers like weight decay. For example, on some simple data sets dropout training produces negative weights even though the output i…

The Lorentz force equations provide a partial description of the geodesic motion of a charged particle on a four-manifold. Under the hypothesis that Maxwell's equations express symmetry properties of the Ricci tensor, the full electromagnetic connection is determined. From this connection, the fourth equation of the ge…

The statistical properties of the increments x(t+T) - x(t) of a financial time series depend on the time resolution T on which the increments are considered. A non-parametric approach is used to study the scale dependence of the empirical distribution of the price increments x(t+T) - x(t) of S&P Index futures, for time…

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 …

Adaptive LR improves neural network Lipschitz regularity without slowing convergence.

We study the Cauchy problem for the wave equation on extreme Kerr backgrounds under axisymmetry. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface S which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain …

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

In this paper, we study the volume growth property of a non-compact complete Riemannian manifold $X$. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to study gradient Ricci solitons. We also show that on $X$, for any $q\in (0,\infty)$,…

Extends Minkowski stability proof to minimal decay assumptions.

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