Paper shows certain algebra types are not differentially smooth.
problem Characterizing smoothness in double extension regular algebras.
method Analyzing algebra type (14641) for differential smoothness.
result Double extension regular algebras of type (14641) are not differentially smooth.
The study examines differential smoothness in specific Artin-Schelter regular algebras of dimension 5.
problem Investigating the differential smoothness of Artin-Schelter regular algebras of dimension 5.
method Analyzing the relationship between the number of generators and Gelfand-Kirillov dimension to identify structural obstructions.
result Certain two- and four-generator AS-regular algebras of global dimension five fail to admit a differential calculus, while a five-generator graded Clifford algebra provides a positive example.
Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisi…
Regularization improves robustness of smoothed classifiers.
problem Certifying robustness of smoothed classifiers.
method Regularizing prediction consistency over Gaussian noise.
result Significantly improved certified robustness with less training costs.
Generic smooth boundaries for isoperimetric regions in 8D manifolds.
problem Understanding boundaries of isoperimetric regions in high-dimensional spaces.
method Generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight.
result Smooth nondegenerate boundaries for isoperimetric regions for generic metrics and volumes.
New method smooths optimization for sparse regularization.
problem Non-smooth, non-convex optimization problems for sparsity.
method Overparameterization and smooth surrogate penalties.
result Surrogate objective has identical global and local minima.
Introduces non-regular spacetime geometry without smooth calculus.
problem Defining gravity without smooth spacetime geometry.
method Discusses non-regular spacetime geometry and curvature without differential calculus.
result Curvature and gravity can be defined without smooth spacetime calculus.
Study evolutes of curves with varying smoothness.
problem Understanding evolutes of curves with low smoothness.
method Analyzing the relationship between curve smoothness and evolute regularity.
result Evolutes have one less order of smoothness than the parent curve in generic cases.
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
Introduces Lie group actions in smoothing processes for currents and spaces with curvature.
problem Regularization of currents and metrics on manifolds and spaces with curvature.
method Actions of compact Lie groups in De Rham approximation and smoothing of Riemannian metrics.
result Effective smoothing processes for currents and metrics on manifolds and spaces with curvature.
A regular n-gon inscribing a knot is a sequence of n points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular n-gon for any n.
Study smooth convergence of metric flows from F-limits.
problem Smooth convergence of F-limit flows. method Extensively studied metric flows and F-limits, showing smooth convergence at regular points. result Each regular point on the limit is a point of smooth convergence.
We investigate the learning rate of multiple kernel learning (MKL) with ℓ1 and elastic-net regularizations. The elastic-net regularization is a composition of an ℓ1-regularizer for inducing the sparsity and an ℓ2-regularizer for controlling the smoothness. We focus on a sparse setting where the total …
Paper introduces a new regularization method for kernel gradient descent learning.
problem Preventing overfitting in kernel gradient descent learning.
method Random smoothing regularization as novel convolution-based smoothing kernels.
result Optimal convergence rates achieved in various function spaces.
New method improves robustness of smoothed classifiers.
problem Improving accuracy and robustness of smoothed classifiers.
method Regularized risk with adaptive regularization.
result Tighter robustness bounds with high probability.
Convex solutions to a specific equation are smooth when the phase is smooth enough.
problem Regularity of solutions to the Lagrangian mean curvature equation.
method Showed regularity for convex solutions under Hölder continuity conditions on the phase.
result Convex viscosity solutions are regular if the Lagrangian phase is Hölder continuous.
New framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.
problem Comparative analysis of regularization norms in ill-posed problems.
method Small noise analysis framework for Tikhonov and RKHS regularizations.
result Optimal convergence rates achieved with adaptive fractional RKHS, but hyper-parameters decay too fast.
New iterative regularization method tackles non-smooth, non-strongly convex functionals.
problem Tackles non-smooth, non-strongly convex functionals in regularization problems.
method Primal-dual algorithm with convergence and stability analysis.
result First iterative regularization procedure for non-smooth, non-strongly convex functionals.
Smooth Yang-Mills fields proved in supercritical dimensions.
problem Regularity of weak Yang-Mills connections in high dimensions.
method ε-regularity theorem, Coulomb gauges construction.
result Stationary Yang-Mills fields are smooth away from a small singular set.
Modern deep neural networks require a tremendous amount of data to train, often needing hundreds or thousands of labeled examples to learn an effective representation. For these networks to work with less data, more structure must be built into their architectures or learned from previous experience. The learned weight…
Study proves solenoidal injectivity for tensor fields on curved manifolds with low regularity.
problem Injectivity for tensor fields on negatively curved manifolds with low regularity metrics.
method Pestov energy estimates for transport equation on non-smooth unit sphere bundle, keeping track of regularity, and using functions with more vertical than horizontal regularity.
result Proves solenoidal injectivity for tensor fields on simple Riemannian manifolds with C1,1 metrics and non-positive sectional curvature. The purpose of this work is to develop and study a distributed strategy for Pareto optimization of an aggregate cost consisting of regularized risks. Each risk is modeled as the expectation of some loss function with unknown probability distribution while the regularizers are assumed deterministic, but are not required…
We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our met…
Smoothness of Hamiltonian stationary submanifolds in symplectic manifolds proven.
problem Smoothness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds.
method Developed a regularity theory for fourth order nonlinear elliptic equations with two distributional derivatives.
result Any C1-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. Establish C^{1,2} regularity of American value functions in Heston model
problem Regularity of American put options in Heston model
method PDE techniques
result C^{1,2} regularity in exercise domain and smooth-fit principle
The study connects group structure to smooth actions on one-manifolds.
problem Understanding how group actions affect the smoothness of manifolds.
method Analyzes the relationship between group algebraic structure and smoothness of group actions on one-dimensional manifolds.
result Uniform construction of groups acting on compact interval and circle with prescribed regularity.
The paper links set cuspidality to function regularity and flatness.
problem Linking set cuspidality to function regularity and flatness.
method Analyzes arc-smooth functions and their properties on various sets.
result Establishes a precise link between set cuspidality and function regularity.
Smooth solutions found for a specific type of Yamabe problem.
problem Regularity of viscosity solutions to the σk-Yamabe problem in the negative cone. method Analysis of Lipschitz viscosity solutions with specific assumptions.
result Existence and smoothness of solutions away from a negligible set.
The paper analyzes the trade-off between smoothness and sparsity in GCN using lp-regularized learning.
problem Quantifying the trade-off between smoothness and sparsity in GCN.
method Proposes a novel SGD proximal algorithm for GCNs with an inexact operator to analyze the stability of the ℓp-regularized stochastic learning. result Establishes an explicit theoretical understanding of GCN with ℓp-regularized stochastic learning. Regularization is an effective way to promote the generalization performance of machine learning models. In this paper, we focus on label smoothing, a form of output distribution regularization that prevents overfitting of a neural network by softening the ground-truth labels in the training data in an attempt to penal…
Paper studies smoothness of bi-conformal heat flow on 4-manifolds.
problem Smoothness of bi-conformal heat flow on 4-manifolds.
method Introduces bi-conformal heat flow (bi-CHF) and proves global smoothness without finite time singularities.
result Global smoothness and no finite time singularity for bi-conformal heat flow.
Smooth low-regular connections lead to smooth immersions with controlled regularity.
problem Smoothability of Lp-connections and existence of isometric immersions with low regularity. method Adapting S. Mardare's work on surface theory, using Hodge decomposition and fixed point theorems.
result Low-regular connections can be approximated by smooth connections of the same curvature.
Regularity results for geodesic X-ray transform on nonsmooth manifolds
problem Geodesic X-ray transform on nonsmooth simple manifolds
method Symbol smoothing arguments and pseudodifferential operators with low regularity symbols
result Improved injectivity results for Lp functions Smooth calibration improves forecast reliability even with leaked information.
problem Improving forecast reliability with leaked information.
method Combining nearby forecasts to ensure smooth calibration, which can be guaranteed by deterministic procedures.
result Smooth calibration can be guaranteed by deterministic procedures even with leaked forecasts, and it yields uncoupled finite-memory dynamics in games.
This paper explores how entropic regularization improves Wasserstein estimators' performance.
problem Improving the approximation and estimation properties of Wasserstein estimators.
method Entropic regularization of optimal transport costs to smooth Wasserstein estimators.
result Entropic regularization can achieve comparable statistical performance to un-regularized estimators at lower computational cost.
Dropout improves regularization in flexible models for rare features.
problem Understanding theoretical properties of dropout in generalized linear models.
method Theoretical analysis and application to adaptive smoothing with B-splines.
result Dropout prefers rare features in mean and dispersion parameters.
Study improves boundary smoothness for area-minimizing currents with complex boundaries.
problem Boundary regularity for area-minimizing currents with arbitrary multiplicity.
method Generalization of Allard's boundary regularity theorem.
result Derivation of structural consequences from the generalized theorem.
Logit regularization induces logit clustering, affecting classifier performance.
problem Understanding the mechanism of logit regularization in classification.
method Analysis of logit regularization in linear classification, proving logit clustering leads to Fisher's Linear Discriminant alignment.
result Logit regularization can halve critical sample complexity and induce robust generalization.
Wasserstein distributionally robust optimization (DRO) has recently achieved empirical success for various applications in operations research and machine learning, owing partly to its regularization effect. Although connection between Wasserstein DRO and regularization has been established in several settings, existin…
The paper proves regularity of states on manifolds with unstable dynamics.
problem Propagation of regularity in dynamical systems with unstable manifolds.
method Leafwise semiclassical pseudodifferential calculus adapted to foliated spaces.
result Pollicott-Ruelle resonant states are smooth over entire manifolds if smooth on unstable leaves.
We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the es…
Smoothness of graphs evolving by fractional mean curvature is proven.
problem Evolution of graphs by fractional mean curvature.
method Analytic semigroup approach to nonlocal quasilinear evolution equation.
result Short time existence, uniqueness, and optimal Hölder regularity of classical solutions.
The study improves off-policy learning by smoothing IPS and provides a generalization bound.
problem Improving off-policy learning from logged bandit data.
method Smooth regularization for IPS, deriving a two-sided PAC-Bayes generalization bound.
result The bound is valid for standard IPS and provides insights into when regularization is useful.
New methods improve convergence in non-convex non-smooth learning problems.
problem Sparse learning from high-dimensional data with non-convex, non-smooth regularizers.
method Stochastic proximal gradient methods with arbitrary sampling.
result Independent sampling improves performance over uniform sampling.
New framework improves sample efficiency and robustness in RL with smooth policies.
problem Sample inefficiency and lack of robustness in deep reinforcement learning.
method SR^2L framework, smoothness-inducing regularization.
result Improved sample efficiency and robustness in both on-policy and off-policy RL algorithms.
Regularized MFPCA smooths multivariate functional data for clearer patterns.
problem Challenges in controlling roughness of multivariate functional PCs.
method ReMFPCA incorporates a roughness penalty in a penalized framework to smooth PCs.
result Smoothed multivariate functional PCs reveal clearer patterns.
Pairwise Label Smoothing improves deep model generalization by reducing overconfidence.
problem Improving deep model generalization through regularization.
method PLS smooths labels for pairs of samples, learning distribution mass during training.
result PLS significantly outperforms LS and baseline models, reducing up to 30% classification error.
Derandomizing PAC-Bayes bounds for smooth loss functions
problem Derandomizing PAC-Bayes bounds for smooth loss functions
method Exploiting smoothness properties of both the loss and the predictor class
result Bounds for deterministic predictors that involve flatness quantities