arXiv research

Last SGD iterate bounds for overparameterized linear regression.

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…

In this paper we investigate the life-span of classical solutions to the hyperbolic geometric flow in two space variables with slow decay initial data. By establishing some new estimates on the solutions of linear wave equations in two space variables, we give a lower bound of the life-span of classical solutions to th…

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

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

We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an exp…

One knows that the large time heat decay exponent on a nilpotent group is given by half the growing rate of the volume of its large balls. This work deals with the similar problem of trying to interpret geometrically the heat decay on (one) forms. We will show how it is (partially) related to the depth of the relations…

New inequality shows energy growth and decay in geometric problems.

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

We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.

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.

Study constructs scattering theory for massless Dirac field on Kerr spacetime.

Survey on stability of Minkowski spacetime in relativity.

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

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

Minimax optimal convergence rates for classes of stochastic convex optimization problems are well characterized, where the majority of results utilize iterate averaged stochastic gradient descent (SGD) with polynomially decaying step sizes. In contrast, SGD's final iterate behavior has received much less attention desp…

The energy in a square membrane $Ω$ subject to constant viscous damping on a subset $ω\subset Ω$ decays exponentially in time as soon as $ω$ satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate $τ(ω)$ of this decay satisfies $τ(ω)= 2 \min(-μ(ω), g(ω))$ (see Lebeau [Math. Phys. Stud. …

New method uses entropy dissipation to prove isoperimetric inequalities.

Geometric focusing affects dispersive estimates for Schrödinger and wave equations.

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…

The paper establishes pressure gaps for manifolds with flat subtori singularities.

We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any polynomial.

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

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…

New exponential decay estimate for Hermitian Yang-Mills metrics near branch points.

Novel heat flow estimates on ALE manifolds for Schrödinger operators.

We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to $-1$ and are $C^0$, but are not necessarily $C^1$, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves a…

We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the complement of a standard tubular neighborhood of the cone-locus whose pointwise bi-Lip…

Establish a unified framework for negative results in Fourier analysis.

We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum and the time of the supremum of a general Lévy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in $L^p$ (for any $p\geq 1$) as a…

The paper establishes curvature estimates for solitons in higher dimensions.

Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is used in the sense that all covariant derivatives of the Riemannian curvature tensor…

Study of 4D Ricci solitons with symmetry, finding precise geometric asymptotics.

Study on efficiency of Dutch auctions on blockchains considering various parameters.

We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in $\mathbb{R}^2$ introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmoni…

Wide neural networks with weight decay exhibit neural collapse.

We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspe…

Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X) when X is a simply connected, nonpositively curved metric spaces by considering…

Self-similar solutions to geometric flows are stable under small perturbations.

Spectral flow connects manifold geometry to rigidity criteria.

S2D selectively decays large singular values to improve quantization of neural activations.

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

In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss-Bonnet curvature is integrable and the decay order $τ$ satisfies $τ> \frac {n-4}{3}.$ Then we show a positive mass theorem for …

Study on spin-zero rest-mass fields using conformal geometric method.

Boosting framework for vector-valued prediction with geometric stability.

Extends Minkowski stability proof to minimal decay assumptions.

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

We prove hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance form the boundary of the convex core for points in the th…