arXiv research

Sharp decay constant for positive scalar curvature metrics on manifolds.

Riemannian stochastic gradient descent converges faster with increasing batch size.

Regularization in the optimization of deep neural networks is often critical to avoid undesirable over-fitting leading to better generalization of model. One of the most popular regularization algorithms is to impose L-2 penalty on the model parameters resulting in the decay of parameters, called weight-decay, and the …

Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space.

New decay estimates for scalar curvature of steady gradient Ricci solitons.

Study on optimal ReLU networks with weight decay for interpolation.

Last SGD iterate bounds for overparameterized linear regression.

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…

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…

Sharp bounds on quasimode norms on compact space forms.

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its $C^2$-distance fro…

We consider the least-squares regression problem and provide a detailed asymptotic analysis of the performance of averaged constant-step-size stochastic gradient descent (a.k.a. least-mean-squares). In the strongly-convex case, we provide an asymptotic expansion up to explicit exponentially decaying terms. Our analysis…

Adaptive LR improves neural network Lipschitz regularity without slowing convergence.

Step decay schedules improve convergence in non-convex optimization.

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…

Paper develops an online learning algorithm for functional data models.

The study examines Kernel Ridge Regression error rates across noiseless and noisy conditions.

The study examines averages of Laplacian determinants over large genus moduli spaces.

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

In recent work, we have proven uniform decay bounds for solutions of the wave equation $\Box_gφ=0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $|φ|\le Cv_+^{-1}$, which holds throughout the domain of outer communications, where $v$ is an advanced Eddington-Finkelstein coordinate, $v_+=\ma…

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 …

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…

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

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

Growth of spinors in 4D and 3D generalized Seiberg-Witten equations.

WSqD extends learning rate schedules for large model training without fixed horizons.

Future stability of FLRW solutions in expanding 3D space is shown for compact perturbations.

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

Paper extends foliation results in higher dimensions for Schwarzschild spaces.

We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \cite{TVMOD}. For the pr…

CNN layers with large norms are still robust to adversarial attacks.

Gradient descent outperforms ridge regression under certain covariance matrix decay conditions.

Study on U-statistics with heavy-tailed samples, providing tail bounds and LDP.

We show that the only complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of the standard ones. This gives a new proof of the Hamilton-Ivey-Perel'man classification of 3-dimensional shrinking gradient solitons. We also prove a classification for expanding gradient Ricci solitons with con…

The concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the Laplacian. In particular, this paper discusses the question of whether there is an eigenvalue of the Laplacian embedded in the essential spectrum or not. The borderline-behavior of the radial curv…

Generalizes Ricci flow starting from small curvature concentration with a Morrey-type condition.

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

New framework for Adam-type algorithms with constant β1, improving regret analysis.

We show that for an $n$ dimensional complete non Ricci flat gradient steady Ricci soliton with potential function $f$ bounded above by a constant and curvature tensor $Rm$ satisfying $\overline{\lim}_{r\to \infty} r|Rm|<\frac{1}{5}$, then $|Rm|\leq Ce^{-r}$ for some constant $C>0$, improving a result of [36]. For any f…

We consider solutions to the linear wave equation $\Box_gφ=0$ on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may con…

Stock prices are observed to be random walks in time despite a strong, long term memory in the signs of trades (buys or sells). Lillo and Farmer have recently suggested that these correlations are compensated by opposite long ranged fluctuations in liquidity, with an otherwise permanent market impact, challenging the s…

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

A 3D metric conformally related to Arnold cat fast dynamo metric: ${ds_{A}}^{2}=e^{-λz}dp^{2}+e^{λz}dq^{2}+dz^{2}$ is shown to present a behaviour of non-dynamos where the magnetic field exponentially decay in time. The Riemann-Christoffel connection and Riemann curvature tensor for the Arnold and its conformal counter…

Let 0<m<(n-2)/n, n>2, $α=(2β+ρ)/(1-m)$ and $β>mρ/(n-2-mn)$ for some constant $ρ>0$. Suppose v is a radially symmetric symmetric solution of $\frac{n-1}{m}Δv^m+αv+βx\cdot\nabla v=0$, v>0, in $R^n$. When m=(n-2)/(n+2), the metric $g=v^{4/(n+2)}dx^2$ corresponds to a locally conformally flat Yamabe shrinking gradient soli…

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

We have created a framework for analyzing subscription based businesses in terms of a unified metric which we call SCV (single customer value). The major advance in this paper is to model customer churn as an exponential decay variable, which directly follows from experimental data relating to subscription based busine…

A new flow method reduces Lorentz contraction to a simple algebraic decay.

We make some improvements to our previous results. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. These results al…