Study geodesic X-ray transform on curved spaces, proving injectivity for decaying functions.
problem Injectivity of geodesic X-ray transform on curved spaces.
method Proving injectivity for decaying functions and tensor fields of any order.
result Injectivity of the geodesic X-ray transform for functions and tensor fields of any order on Cartan-Hadamard manifolds.
The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.
problem Improving energy decay estimates for Dir-stationary Q-valued functions.
method Establishing improved decay estimates and applying them to derive Liouville-type theorems and continuity.
result Dir-stationary Q-valued functions exhibit the Lebesgue property and reside in a generalized Campanato-Morrey space.
The paper extends decay estimates to graphs with positive spectrum.
problem Proving decay estimates for nonnegative functions on graphs.
method Sharp ℓ2 decay estimates for nonnegative generalized subharmonic functions. result Extends Li and Wang's result to graphs with positive Laplacian spectrum.
Study on curvature decay in steady Ricci solitons, proving dichotomy.
problem Curvature decay in steady Ricci solitons.
method Established a dichotomy for curvature decay in specific types of solitons.
result Proved a dichotomy on curvature decay for certain steady Ricci solitons.
Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.
problem Analyzing properties of Kähler manifolds with nonnegative bisectional curvature.
method Established precise relations among minimal degree, volume growth, and scalar curvature decay.
result Unified understanding of Kähler-Ricci flow through polynomial growth holomorphic functions.
Study sharp decay of capacity for subharmonic functions on compact Hermitian manifolds.
problem Sharp decay of capacity of sublevel sets of (ω,m)-subharmonic functions. method Generalizes previous results on Kähler manifolds and obtains full characterizations of polar sets.
result Full characterizations of polar sets and extremal functions.
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…
Optimal learning rates decay to zero in easy tasks and maintain a warmup phase in hard tasks.
problem Optimizing learning rates under functional scaling laws for model training.
method Deriving optimal learning-rate schedules based on exponents s and β. result Sharp phase transition between easy and hard tasks, with different decay behaviors.
Study on optimal ReLU networks with weight decay for interpolation.
problem Interpolating data with radially symmetric distributions using shallow ReLU networks.
method Weight decay regularization in infinite neuron, infinite data limit; analysis of growth rates.
result Existence and growth rates of unique radially symmetric minimizers with weight decay.
In this paper we study potential function of gradient steady Ricci solitons. We prove that infimum of potential function decays linearly; in particular, potential function of rectifiable gradient steady Ricci solitons decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential…
Study shows decay of correlations on specific types of flows.
problem Analyzing decay of correlations in specific flow types.
method Asymptotic expansion of correlation function on Abelian covers.
result Established an expansion in inverse powers of time.
Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
problem Understanding the rigidity of Ricci-flat manifolds with specific curvature decay.
method Analyzing the gradient of the Green function and using curvature decay conditions.
result Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
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 …
Study on optimal rates for learning algorithms with polynomial eigenvalue decay.
problem Understanding convergence rates of learning algorithms under general source conditions.
method Analyzes Tikhonov regularization and operator monotone index functions in minimax setting.
result Establishes upper convergence rates and minimum possible error for learning algorithms.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
Paper calculates eigenvalue decay rates for neural network kernels on general domains.
problem Determining eigenvalue decay rates for neural network kernels on arbitrary domains.
method Proved dynamics of wide neural networks approximates NTK on general domains, used minimax optimality and interpolation spaces.
result Provided strategy to calculate eigenvalue decay rates for neural network kernels.
Geometric step decay schedules improve stochastic algorithms' convergence on sharp nonconvex problems.
problem Convergence of stochastic algorithms on sharp nonconvex problems.
method Geometric step decay schedule applied to stochastic algorithms.
result Geometric step decay schedules lead to local linear convergence rates for sharp nonconvex problems.
Paper shows existence of vortex solutions with specific decay properties.
problem Existence of solutions to Seiberg-Witten equations with specific decay properties.
method Dimensional reduction of Seiberg-Witten equations on the plane.
result Contains both exponentially decayed and polynomial growth solutions.
The paper studies Teichmüller TQFT for hyperbolic knots, proving exponential decay of partition functions.
problem Analyzing Teichmüller TQFT for hyperbolic knots with generalized FAMED triangulations.
method Introducing generalized FAMED property, proving exponential decay of partition functions in semi-classical limit.
result Partition functions decay exponentially with the volume of knot complements, and the 1-loop invariant emerges.
Adaptive time decay functions improve financial product recommendation accuracy.
problem Inaccurate recommendations due to static historical data in finance.
method Time-dependent collaborative filtering with personalized decay functions.
result Significant improvements over state-of-the-art benchmarks in financial product recommendation.
Weight decay stabilizes training dynamics by slowing progressive sharpening.
problem Understanding how weight decay affects training stability in deep learning models.
method Analyzing weight decay effects at the Edge of Stability, developing a mathematical framework.
result Weight decay dampens oscillations and stabilizes sharpness in CNNs, causing a phase transition in MLPs.
Introduces gradient decay in Softmax for better generalization.
problem Improving generalization performance in neural networks.
method Gradient decay hyperparameter in Softmax for varying gradient rates based on probability.
result Gradient decay rate affects generalization performance and can be tuned for better optimization.
Investigates kernel regression rates without assuming polynomial eigenvalue decay.
problem Achieving minimax rates without strict assumptions on kernel eigenvalue decay.
method Examines kernel regularization methods under weak assumptions on eigenvalue decay.
result Achieves minimax convergence rates under less restrictive conditions.
Regularizers change the geometric properties of loss functions in neural networks.
problem Understanding how different regularizers affect the geometric properties of loss functions in neural networks.
method Examined several regularizers, including weight decay, to determine if the regularized loss function becomes Morse.
result For certain regularizers, the regularized loss function becomes Morse, indicating a change in geometric properties.
Study finds solutions to inequality decay to zero on warped cylinders.
problem Analyzing solutions to a specific inequality on warped cylindrical ends.
method New Carleman estimate for independent interest.
result Solutions decay to zero along warped cylindrical ends.
The Weyl curvature in expanding black hole cosmologies decays towards infinity.
problem Non-linear stability of expanding regions in Kerr de Sitter cosmologies.
method Establishes decay estimates for Weyl fields under realistic assumptions.
result Uniform, optimal, and consistent decay estimates for Weyl fields.
Develops heat kernel and Green's function estimates for manifolds.
problem Solving Poisson equation on manifolds with Ricci curvature bounds.
method Heat kernel and Green's function estimates for manifolds with positive spectrum.
result Existence and sharp estimates of Poisson equation solutions on manifolds with Ricci curvature bounds.
We found that factors decay over time, with momentum fitting best.
problem Understanding how factors decay over time and their impact on performance.
method Derived a hyperbolic decay model for factors, tested against linear and exponential alternatives.
result Momentum exhibits hyperbolic decay, outperforming linear and exponential models.
Adaptive weight-decay improves deep neural network performance.
problem Overfitting in deep neural networks.
method AdaDecay adjusts weight-decay adaptively based on gradient norms within each layer.
result AdaDecay improves generalization and accuracy across various datasets and models.
New proof removes decay assumptions for spacetime positive mass theorem.
problem Proving the rigidity of the spacetime positive mass theorem without additional decay assumptions.
method Uses spacetime harmonic functions and Liouville's theorem, and an alternative proof based on Killing development.
result Removes additional decay assumptions for the spacetime positive mass theorem.
Paper develops an online learning algorithm for functional data models.
problem Recovering slope functions or predictors in functional data models.
method Online regularized learning algorithm in reproducing kernel Hilbert spaces with polynomially decaying step-size.
result Established fast convergence rates for estimation error without capacity assumption.
Weight decay outperforms adversarial training for robustness.
problem Improving machine learning model robustness to adversarial attacks.
method Weight decay vs adversarial training for robustness.
result Weight decay is superior to adversarial training in terms of robustness.
The paper proves scalar curvature decay for uniformly contractible manifolds with finite asymptotic dimension.
problem Proving decay of scalar curvature for uniformly contractible manifolds with finite asymptotic dimension.
method Using index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control, and Lipschitz control for topological K-theory of finite dimensional simplicial complexes.
result The scalar curvature decays to zero at a rate depending only on the contractibility radius and the diameter control of the asymptotic dimension.
SignSGD outperforms SGD in linear regression with optimal scaling laws under PLRF model.
problem Improving linear regression performance with signSGD under power-law random features.
method Analysis of signSGD risk under PLRF model, comparison with SGD, identification of unique effects.
result SignSGD can have a steeper compute-optimal slope than SGD in noisy regimes, especially with WSD schedule.
Study examines implied volatility behavior in Bachelier model.
problem Characterizing implied volatility in Bachelier model for large strikes.
method Exploiting regular variation theory, derived explicit expressions for Bachelier implied volatility.
result Established a rigorous connection between characteristic function analyticity and volatility smile asymptotic slope.
Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi-martingale models. It is shown that as the position size approaches infinity, the utility function's decay rate for large negative…
Study shows slow decay of impact in equity markets after metaorders.
problem Understanding the decay of impact in equity markets after metaorders.
method Empirical study using a large dataset of metaorders executed by institutional investors.
result Impact decay follows a power-law function at short time scales and converges to a non-zero asymptotic value at long time scales.
The paper establishes curvature estimates for solitons in higher dimensions.
problem Curvature estimates for steady and expanding solitons in higher dimensions.
method Curvature estimates using gradient Ricci solitons and integral estimates.
result Curvature operator decays at specific rates for different cases of solitons.
We discover scaling laws for kernel regression loss under various learning rate schedules.
problem Understanding loss dynamics and learning rate schedules in kernel regression.
method Theoretical analysis of stochastic gradient descent on a power-law kernel regression model.
result Established a Functional Scaling Law (FSL) capturing the full loss trajectory under arbitrary learning rate schedules.
Study of flows on circle bundles over translation surfaces, showing decay of correlations.
problem Ergodic properties of flows on circle bundles over translation surfaces.
method Generalizing Heisenberg nilflows to more general base surfaces, showing relatively mixing.
result Showed that such flows exhibit decay of correlations in the orthogonal complement of functions constant along fibers.
Improved learning rate schedule for least squares regression.
problem Achieving optimal convergence rates for least squares regression.
method Step Decay schedule with geometrically decaying learning rates.
result Final iterate behavior with Step Decay schedules is off the minimax rate by only log factors.
The paper constructs Ricci-flat Kähler manifolds with specific decay properties.
problem Finding Ricci-flat Kähler manifolds with controlled decay rates.
method Geometric existence proof and construction of ansatz.
result Existence of 39 distinct Ricci-flat Kähler 3-folds with specific asymptotic angles.
In this paper we continue our study of the Laplacian on manifolds with axial analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using the complex scaling method and the Phragmén-Lindelöf principle we prove exponential decay of the eigenfunctions corresponding to the non-threshold eigenvalues of t…
Paper addresses bias in kernel density estimation under minimal assumptions.
problem Kernel density estimation bias under minimal assumptions.
method Demonstrates the need for a balance between kernel decay and bandwidth eigenvalues, and rigorously derives bias bounds.
result Explicit constants and rigorous derivation of bias bounds under minimal assumptions.
Classifies scalar-flat toric Kähler instantons in 4D.
problem Classifying scalar-flat toric Kähler 4-manifolds.
method Using Liouville theorem for degenerate-elliptic equations, classifies momentum functions and metrics.
result Fully classifies instantons with ALE-F-G-H asymptotic types.
We study the tick dynamical behavior of the bond futures in Korean Futures Exchange(KOFEX) market. Since the survival probability in the continuous-time random walk theory is applied to the bond futures transaction, the form of the decay function in our bond futures model is discussed from two kinds of Korean Treasury …
Deep ReLU networks can approximate various signal types with exponential error decay.
problem Approximating different signal structures with deep neural networks.
method Demonstrated approximation of polynomials, sinusoidal functions, oscillatory textures, and fractals.
result Finite-width deep ReLU networks require fewer connections than wide finite-depth networks for smooth function approximation.
Active data collection improves convergence rates in operator learning.
problem Improving convergence rates in operator learning with linear target and stochastic input.
method Active data collection strategies with mean-zero stochastic process and continuous covariance kernels.
result Achieves arbitrarily fast error convergence rates with eigenvalue decay of covariance kernels.