Study proves uniqueness of asymptotic limits for specific manifolds.
problem Proving uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth.
method Established using natural curvature and cross section assumptions.
result Uniqueness and exponential convergence rate for complete noncollapsed Ricci-flat manifolds with linear volume growth.
Establishes geometric convergence of iterative optimization algorithms.
problem Analyzes convergence of iterative optimization algorithms under general assumptions.
method General framework for iterative optimization algorithms, proving asymptotic geometric convergence and providing convergence rates.
result Asymptotic geometric convergence of iterative optimization algorithms with exact rate.
Unified approach to compute asymptotic constants using optimization.
problem Computing unknown constants in asymptotic expansions.
method Linear Least Squares and Tikhonov Linear Least Squares methods.
result Rigorous asymptotic estimates and convergence-rate guarantees.
The paper proves a new theorem about paths on nilmanifolds.
problem Understanding the asymptotic behavior of paths on nilmanifolds.
method Analytic mechanism involving uniform convergence and layer-by-layer convergence of nilpotent developments.
result Recovering and extending previous theories to arbitrary nilpotent steps.
In this paper we define a new convergence called "asymptotically conic convergence" in which a smooth family of Riemannian metrics on a fixed compact manifold degenerate to a metric with isolated conic singularity. Our results are: convergence of the spectrum of the geometric Laplacians and uniform convergence of the c…
The Yamabe flow on flat manifolds converges to a scalar flat metric.
problem Analyzing the convergence of Yamabe flow on asymptotically flat manifolds.
method Yamabe flow starting from an asymptotically flat manifold, convergence analysis.
result The flow converges to an asymptotically flat, scalar flat metric under certain conditions.
Ricci flow on flat manifolds converges to Euclidean space under curvature pinching.
problem Curvature pinching on asymptotically flat manifolds.
method Ricci flow on asymptotically flat manifolds with integral curvature pinching.
result Ricci flow converges to flat Euclidean space for sufficiently pinched curvature.
Random forests' performance is analyzed with rates of convergence and asymptotic normality established.
problem Theoretical understanding of random forests' performance and behavior.
method Generalized U-statistics framework to analyze random forest predictions.
result Random forest predictions can remain asymptotically normal for larger subsample sizes.
New quasi-Newton method guarantees global superlinear convergence.
problem Global convergence and superlinear convergence of quasi-Newton methods.
method Hybrid proximal extragradient method with online learning for Hessian approximation.
result First globally convergent quasi-Newton method with explicit superlinear convergence rate.
Improved TD learning for non-i.i.d. Markovian data.
problem Convergence analysis of two time-scale TD learning under Markovian samples.
method Non-asymptotic convergence analysis of two time-scale TD with gradient correction under Markovian data.
result Two time-scale TD can converge as fast as O(log t/(t^(2/3))) under diminishing stepsize.
Study shows how fast a specific matrix completion method works.
problem Completing a rank-one matrix from a subset of revealed entries.
method Alternating minimization approach for matrix completion.
result Polynomial upper bound on convergence rate.
The Yamabe flow converges to a specific function on compactified manifolds.
problem Analyzing the Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant.
method Studied the Yamabe flow on asymptotically flat manifolds with Y≤0 and showed convergence after rescalings. result The Yamabe flow converges to the unique positive function solving the Yamabe problem on a compactification of the original manifold.
Curve shortening flow converges to a translating soliton for certain curves.
problem Asymptotic behavior of curve shortening flow.
method α-curve shortening flow for exponents α > 1/2.
result Converges to the unique translating soliton.
New classifiers converge under large data, simplifying complex models.
problem Complex predictive models under large datasets.
method Convergence of simultaneous and marginal classifiers under partition exchangeability.
result Asymptotic convergence of classifiers with large data reduces computational complexity.
Flow of curved surfaces converges to a specific shape over time.
problem Behavior of curved surfaces over time.
method Addressed through the α-Gauss curvature flow for α>1/2. result Flow converges to a translating soliton determined by initial conditions.
The paper proves uniqueness and convergence of asymptotically conical self-shrinkers and self-expanders.
problem Proving uniqueness and convergence of asymptotically conical self-shrinkers and self-expanders.
method Analyzing properly immersed mean curvature flow self-shrinkers and self-expanders asymptotic to cones.
result Proves uniqueness and convergence of asymptotically conical self-shrinkers and self-expanders.
Develops a generalized version of Chung's Lemma for stochastic optimization methods.
problem Establishing asymptotic convergence rates for stochastic optimization methods under various step size rules.
method Generalized version of Chung's Lemma for a broader family of step size rules.
result Demonstrates tight non-asymptotic convergence rates for various stochastic methods.
Study geodesic Lie groups' convergence to limits with quantitative estimates.
problem Quantifying convergence rates of geodesic Lie groups to their limits.
method Estimates on the difference between original metrics and asymptotic/tangent metrics.
result Sharpens existing bounds on convergence rates.
Continuous-time distributed mirror descent with integral feedback converges to global optimum.
problem Distributed optimization of a global strongly convex function with local convex components.
method Continuous-time distributed mirror descent with integral feedback.
result Asymptotic convergence to global optimum with constant step-size.
Lower semicontinuity of mass in 3D asymptotically flat manifolds proven.
problem Lower semicontinuity of mass in asymptotically flat 3-manifolds.
method Used Huisken's isoperimetric mass and modified weak mean curvature flow.
result Total mass is lower semicontinuous under C0 convergence. Study shows how flat flow solutions in 2D converge to disks.
problem Understanding the asymptotics of area-preserving mean curvature flow in 2D.
method Analyzes flat flow solutions starting from bounded sets of finite perimeter.
result Flat flow solutions converge to a union of equally sized disks with exponential rate.
Gradient flows of neural networks converge to optimal values or diverge, with thresholds and asymptotic behaviors.
problem Understanding the convergence and divergence of gradient flows in neural networks.
method Analysis of gradient flows on loss landscapes of neural networks using o-minimal structures.
result Gradient flows either converge to optimal values or diverge to infinity, with thresholds and asymptotic behaviors.
We characterize sequences of Kleinian surface groups with convergent subsequences in terms of the asymptotic behavior of the ending invariants of the associated hyperbolic 3-manifolds. Asymptotic behavior of end invariants in a convergent sequence predicts the parabolic locus of the algebraic limit as well as how the a…
This study analyzes AdaGrad's stability and convergence in non-convex optimization.
problem Lack of theoretical analysis for AdaGrad in non-convex optimization.
method Novel stopping time-based techniques from probability theory.
result Established stability and derived convergence rates for AdaGrad.
This work analyzes DP-SGD for online LDP problems with practical convergence rates.
problem Analyzing DP-SGD for online LDP problems with practical convergence rates.
method Developed a general framework for online LDP model in stochastic optimization problems, conducted non-asymptotic convergence analysis.
result Comprehensive non-asymptotic convergence analysis of the proposed estimators in finite-sample situations.
Study of 2+1 dimensional cosmologies with positive cosmological constant, proving asymptotic convergence to de Sitter.
problem Asymptotic behavior of 2+1 dimensional cosmologies with positive cosmological constant.
method Mean Curvature Flow methods.
result Spatial slices asymptotically converge to de Sitter, becoming physically indistinguishable from it.
Paper derives convergence rates and confidence intervals for LSA with Markovian noise.
problem Analyzing convergence rates and constructing confidence intervals for LSA with Markovian noise.
method Derives non-asymptotic Berry-Esseen bounds and multiplier block bootstrap procedure.
result Provides O(n−1/4) convergence rates and guarantees consistent inference. New method shows Hessian estimator from random samples converges to true Hessian on complex manifolds.
problem Uncertainty in Hessian estimator accuracy on complex manifolds with boundaries and nonuniform sampling.
method Locally fitting quadratic polynomials, rigorous theoretical analysis under mild conditions.
result The Hessian estimator asymptotically converges to the true Hessian, even near boundaries.
Uniform counting formulas for orthogeodesics in Kleinian groups converge.
problem Counting orthogeodesics in Kleinian groups converging to a limit.
method Spectral gap of the limit manifold and geodesic flow mixing property.
result Asymptotically uniform counting formulas for orthogeodesics.
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an L2-gradient flow for Euler's elastic energy. result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
CR method improves convergence for nonconvex optimization under KL property.
problem Improving convergence rate for nonconvex optimization problems.
method Cubic-regularized Newton's method exploiting Kurdyka-Lojasiewicz (KL) property.
result Asymptotic convergence rates of various optimality measures are fully characterized.
Study shows spherical hyperbolic manifolds almost rigidly converge to hyperbolic space.
problem Almost rigidity of positive mass theorem for spherical hyperbolic manifolds.
method Intrinsic flat distance to prove convergence.
result Spherically symmetric asymptotically hyperbolic manifolds converge to hyperbolic space if mass limit is zero.
The paper shows how to stabilize perturbed Kähler-Ricci solitons.
problem Stabilizing perturbed Kähler-Ricci solitons.
method Normalized Kähler-Ricci flow starting from perturbed metrics.
result The flow converges to an asymptotically conical gradient expanding Kähler-Ricci soliton.
Improves understanding of stochastic NGVI convergence rates.
problem Lack of knowledge about non-asymptotic convergence rates in stochastic NGVI.
method Proved non-asymptotic convergence rates for conjugate likelihoods and showed implicit optimization for non-conjugate likelihoods.
result First O(T1) non-asymptotic convergence rate for stochastic NGVI in conjugate likelihoods. We introduce a natural definition of Lp-convergence of maps, p≥1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of …
Paper explores weighted averaging schemes for SGD, achieving asymptotic normality and optimality.
problem Improving convergence of SGD in various settings.
method Develops a general weighted averaging scheme for SGD and establishes asymptotic normality.
result Establishes asymptotic normality and optimality of weighted averaged SGD solutions.
Flow of spacelike hypersurfaces converges to flat slice in asymptotically flat spacetimes.
problem Long-time behavior of mean curvature flow in asymptotically flat spacetimes.
method Analysis of mean curvature flow in Lorentzian product manifolds.
result Mean curvature flow converges uniformly to a flat slice as time goes to infinity.
The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin-Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic n…
Proofs high-dimensional spectrum convergence of weighted sample covariance.
problem High-dimensional spectrum convergence of weighted sample covariance.
method Proposes a new, concise proof with stronger assumptions.
result Spectrum convergence proven for different weight distributions.
The paper studies heat flow for p-pseudoharmonic maps under curvature constraints.
problem Global existence and asymptotic convergence of p-pseudoharmonic map heat flow.
method Heat flow for p-pseudoharmonic maps from a closed Sasakian manifold to a compact Riemannian manifold.
result Global existence and asymptotic convergence of the solution for the p-pseudoharmonic map heat flow under curvature constraints.
Paper shows robust estimators converge to true risk minimizers at optimal rates.
problem Understanding asymptotic properties of robust risk minimizers.
method Investigates robust analogues of empirical risk minimization, focusing on median of means estimator.
result Robust minimizers converge to true minimizers at optimal rates and have similar asymptotic variance.
We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow …
Paper provides exponential convergence guarantees for Iterative Markovian Fitting.
problem Addressing the Schrödinger Bridge problem in computational optimal transport and generative modeling.
method Develops non-asymptotic exponential convergence guarantees for Iterative Markovian Fitting.
result First non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions.
Study shows smooth convergence of round surfaces in flat space-time models.
problem Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces.
method Volume preserving mean curvature flow in asymptotically flat 3-manifolds.
result The flow converges smoothly to a stable CMC surface.
Study guarantees convergence of mean shift mode estimation.
problem Ensuring reliable mode estimation in KDE using mean shift.
method Utilizes Łojasiewicz inequality to prove convergence rate.
result Extends convergence guarantees to biweight kernel.
Study stability of mass theorems for hyperbolic manifolds foliated by IMCF.
problem Stability of Positive Mass Theorem and Riemannian Penrose Inequality in asymptotically hyperbolic manifolds.
method Sequence of regions foliated by IMCF, convergence to hyperbolic or AdS-Schwarzschild metrics.
result Convergence of regions to specific metrics under given conditions.
Two masses on surfaces with boundary converge to ADM mass.
problem Evaluating quasi-local masses on surfaces with boundaries.
method Hawking mass and Huisken's isoperimetric mass on surfaces with boundary, convergence to ADM mass.
result Convergence of Hawking and Huisken's masses to ADM mass.
The paper proves the stability of a flow in Schwarzschild space.
problem Stability of area preserving mean curvature flow in asymptotic Schwarzschild space.
method Demonstrates existence and exponential convergence of the flow for all time.
result The flow converges to a round sphere or a constant mean curvature surface.