Study shows gap between uniform convergence and test error in random feature models.
problem Understanding the gap between uniform convergence and test error in random feature models.
method Analytical expressions for uniform convergence over norm balls, interpolators, and minimum norm interpolator risk derived and proved.
result Uniform convergence over interpolators still gives a non-trivial bound of test error even when classical uniform convergence is vacuous.
Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.
problem Gap between upper and lower bounds on sample complexity for fat-shattering dimension.
method Provided an improved uniform convergence bound.
result Closed the gap between existing upper and lower bounds on sample complexity.
Paper proves discrete uniformizations converge to continuous for surfaces of genus ≥1.
problem Computing uniformizations for surfaces of genus >1.
method Discrete conformality and uniformization on triangle meshes.
result Discrete uniformizations approximate continuous uniformization for closed surfaces of genus ≥1.
This work establishes uniform convergence of subdifferentials in stochastic optimization.
problem Understanding how empirical stationary points approximate population ones in nonsmooth, nonconvex stochastic optimization.
method Reduction principle for weakly convex stochastic objectives, focusing on subgradient convergence.
result Sharp uniform convergence rates for subdifferential mappings in stochastic convex-composite optimization.
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.
Develops uniform convergence guarantees for a broad class of risk functionals in supervised learning.
problem Bounding generalization gaps for various risk functionals beyond the expectation.
method Establishes uniform convergence for Hölder risk functionals, providing guarantees for empirical risk minimization.
result First uniform convergence results for estimating the CDF of loss distributions, applicable to various risk functionals.
We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.
Smooth DNNs mitigate the curse of dimensionality in uniform convergence for various regression tasks.
problem The curse of dimensionality in uniform convergence of ReLU networks.
method Analysis of smoothly activated deep neural networks (smooth DNNs), establishing pseudo-dimension bounds and non-asymptotic approximation guarantees.
result Smooth DNNs achieve non-asymptotic uniform convergence rates across multiple statistical contexts, mitigating the curse of dimensionality.
Uniform convergence of isotopies implies ambient isotopy, aiding knot equivalence.
problem Determining when uniform convergence of isotopies leads to ambient isotopies.
method Using a diagrammatic condition to offload uniform convergence, constructing examples of tame knots.
result Constructing tame knots with countably-many crossings, distinguishing them from wild curves.
Uniform convergence fails to explain deep learning generalization.
problem Understanding why overparameterized deep networks generalize well despite having many parameters.
method Developed and tested various generalization bounds based on uniform convergence.
result Uniform convergence cannot explain generalization in overparameterized deep networks.
Guarantees uniform convergence for square-root Lipschitz losses.
problem Uniform convergence guarantees for square-root Lipschitz losses.
method Using Rademacher complexity and square root of scalar loss function Lipschitz constant.
result Generalizes previous results and handles non-smooth loss functions.
The paper develops a uniform function estimator in RKHS for regression.
problem Reconstructing functions from noisy data at random locations.
method Using reproducing kernel Hilbert spaces and Gaussian random fields.
result The estimator converges uniformly to the conditional expectation.
Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension
problem Convergence of scalar curvature in Kähler-Ricci flow
method Uniform μ μ μ -entropy or uniform Sobolev inequality result Scalar curvature converges to negative Kodaira dimension
The paper characterizes geometric infiniteness for convergence group actions using orbit uniform metrics.
problem Characterizing geometric infiniteness for convergence group actions.
method Introducing orbit uniform metrics and proving properties of discrete orbits.
result Characterization of geometric infiniteness in terms of uncountability of non-conical limit points and existence of escaping sequences.
Proves convergence of gradient Ricci shrinkers with uniform bounds.
problem Compactness and energy concentration in gradient Ricci shrinkers.
method Bubble-tree convergence and local energy analysis.
result No energy concentrates in neck regions, leading to a local diffeomorphism finiteness theorem.
Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
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.
Uniform estimates prove convergence of Chern-Ricci flow on complex surfaces.
problem Proving convergence of Chern-Ricci flow on complex minimal surfaces.
method Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence; surface torsion estimate, uniform total variation bound, Green-weighted L^2 estimate, linear iteration of real Poisson equations.
result Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence for normalized Chern-Ricci flow on complex minimal surfaces.
Deep learning generalizes well despite being overparameterized.
problem Why deep networks generalize well despite fitting training data perfectly.
method Empirical study of training methods and derivation of data-dependent generalization bounds.
result Uniform convergence alone is insufficient for explaining generalization in overparameterized settings.
Paper proves convergence rates for Gaussian kernel ridge regression.
problem Understanding convergence rates for Gaussian kernel ridge regression.
method Establishes polynomial convergence rates for KRR with fixed hyperparameters.
result First polynomial convergence rates for Gaussian kernel ridge regression.
Study investigates uniform convergence of Teichmueller harmonic map flow.
problem Uniform convergence of Teichmueller harmonic map flow.
method Gradient flow of harmonic map energy on surface metrics.
result Uniform convergence of flow to branched minimal immersion.
Study on convergence rate of Bergman metrics on Kähler manifolds.
problem Analyzing convergence rate of Bergman metrics on Kähler manifolds.
method Using Tian's peak section method to show uniform C 1 , α C^{1,α} C 1 , α convergence. result Uniform C 1 , α C^{1,α} C 1 , α convergence of Bergman metrics is demonstrated. Online learning of linear operators between infinite-dimensional spaces is possible but with limitations.
problem Learning linear operators between infinite-dimensional Hilbert spaces in an online setting.
method Online learning approach for linear operators with bounded p p p -Schatten norm, proving impossibility for operator norm. result Separation between online learnability and uniform convergence for bounded linear operators.
New algorithms achieve uniform stability for empirical risk minimization.
problem Designing uniformly stable optimization algorithms for empirical risk minimization.
method Black-box conversion of smooth optimization algorithms and development of Mirror Descent for smooth optimization.
result Optimal algorithms with uniform stability and convergence rates for smooth optimization.
Study uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
problem Uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
method Analysis of random walks on geometric and directed kNN graphs, using concentration tools and differential geometry.
result Uniform convergence of k k k NN Laplacians to diffusion Laplacian, without continuity of transition kernel. Sharp bounds on uniform generalization errors in binary linear classification.
problem Understanding the uniform generalization errors in binary linear classification.
method Isoperimetric arguments, Poincaré and log-Sobolev inequalities for joint distributions.
result Sharp concentration bounds on uniform generalization errors, almost sure convergence in broad settings.
New method confirms Ricci iteration converges to Kähler-Einstein metrics.
problem Confirming the conjecture that the Ricci iteration converges to Kähler-Einstein metrics.
method Using Perelman's convergence theory for the Ricci flow, the article confirms the conjecture for the Ricci iteration.
result The Ricci iteration converges to Kähler-Einstein metrics, providing a new method of uniformization of the Riemann sphere.
The study examines sequences of Riemannian manifolds with uniform Sobolev bounds and their convergence properties.
problem Analyzing sequences of compact Riemannian manifolds with uniform Sobolev bounds and their convergence.
method Establishing a general trace inequality on Riemannian manifolds and proving convergence properties using Hölder and Intrinsic Flat metrics.
result Sequences of compact Riemannian manifolds with uniform Hölder bounds on their distance functions have subsequences converging in the Gromov--Hausdorff sense.
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 paper studies invariant weighted Bergman metrics on domains.
problem Investigating invariant weighted Bergman metrics under biholomorphisms.
method Introducing invariant weight assignments, using Bergman's minimum integral method and domain version of Tian-Yau-Zelditch expansion.
result Uniform convergence of weighted Bergman kernels and metrics on uniform squeezing domains.
The paper explores why a specific type of predictor works well in noisy data.
problem Understanding why a specific type of predictor (minimum-norm interpolator) works well in noisy data.
method The paper uses uniform convergence and zero-error predictors in a norm ball to explain the success of the minimum-norm interpolator.
result The minimum-norm interpolator is consistent, and this can be explained by uniform convergence of zero-error predictors in a norm ball.
The paper introduces new curvature flows and uniformization theorems for polyhedral surfaces.
problem Discrete uniformization and rigidity of polyhedral surfaces.
method Parameterized discrete curvature, uniformization theorem, Yamabe flow, Calabi flow.
result The flows converge to metrics with constant discrete curvature, confirming conjectures.
Two algorithms converge to dictionary learning with geometric rate for non-uniform data.
problem Dictionary learning for non-uniform data models.
method Derivation of convergence conditions for MOD and ODL.
result Both algorithms converge to the generating dictionary with geometric rate under certain conditions.
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
New sampler improves uniform sampling over convex bodies with fewer queries.
problem Improving uniform sampling over convex bodies with fewer queries.
method Proximal sampler with uniform ergodicity and annealing scheme.
result Converges in Rényi-infinity divergence with O ~ ( d 3 e x t p o l y l o g 1 ε ) \widetilde{\mathcal{O}}(d^3\, ext{polylog} \frac{1}{\varepsilon}) O ( d 3 e x t p o l y l o g ε 1 ) query complexity. New algorithm FLUTE achieves uniform-PAC convergence in RL with linear approx.
problem RL with linear function approximation lacks uniform-PAC guarantees.
method FLUTE algorithm with minimax value function estimator and multi-level partition scheme.
result Uniform-PAC convergence to optimal policy with high probability.
Batch normalization biases linear models towards uniform margins, improving performance in binary classification.
problem Understanding the implicit bias of batch normalization in linear models and neural networks.
method Analyzing gradient descent convergence on linear models and two-layer CNNs with batch normalization.
result Gradient descent with batch normalization in linear models converges to a uniform margin classifier with an exponential convergence rate.
Warped tori with almost non-negative scalar curvature converge to a flat torus.
problem Understanding the behavior of warped product metrics on a 3-torus.
method Analyzing sequences of warped product metrics with specific curvature and volume bounds.
result A subsequence of warped product metrics converges to a flat torus.
The paper discusses methods to compute Green's function on algebraic surfaces using Schottky uniformization.
problem Computing Green's function on algebraic surfaces using Schottky uniformization.
method Investigates convergence of deformations of a formula related to Green's function.
result Provides insights into the geometric interpretation of the formula for Green's function.
New bounds show multicalibration error is close to prediction error.
problem Addressing fairness in machine learning systems.
method Sample complexity bounds for uniform convergence of multicalibration error.
result Uniform convergence guarantees for multicalibration error, independent of prediction error.
Uniform convergence of metrics on vortex moduli space in Bradlow limit.
problem Understanding the geometry of vortex moduli spaces.
method Proof of uniform convergence of metrics using normalized L 2 L^2 L 2 metric and Fubini-Study metric. result Establishes the Fubini-Study metric as the limit of the normalized L 2 L^2 L 2 metric in the Bradlow limit. New method speeds up optimization with non-uniform sampling.
problem Optimizing complex stochastic problems efficiently.
method Stochastic Primal Dual Coordinate Method with Optimality Violation-based Sampling.
result The proposed method and variants outperform other methods in speed.
The study examines sequences of 3D generalised monopoles with uniform bounds.
problem Analyzing sequences of 3D generalised monopoles with uniform L2-norm bounds.
method Focus on Swann bundles and uniform lower bounds on hyperKahler potential.
result Convergent subsequences of generalised monopoles over compact subsets of Y' are found.
Increasing iterate averaging improves convergence rates for saddle-point problems.
problem Solving saddle-point problems efficiently.
method Increasing iterate averaging schemes applied to various first-order methods.
result Increasing iterate averaging preserves the O ( 1 / T ) O(1/T) O ( 1/ T ) convergence rate with no additional assumptions or overhead. The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
problem Proving the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
method Using the ∂ ∂ ‾ \partial\overline\partial ∂ ∂ -class, the study proves the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces. result Uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces, with smooth convergence and bounded curvature for initial metrics in the ∂ ∂ ‾ \partial\overline\partial ∂ ∂ -class of the Tricerri/Vaisman metric. Study Betti numbers of manifolds converging to covers, deriving new convergence results.
problem Analyzing Betti numbers of manifolds converging to their covers.
method Benjamini-Schramm convergence, Price inequalities, refined Thick-Thin decomposition, Moser iteration.
result Convergence results for weakly uniform discrete sequences of closed Riemannian manifolds under negative Ricci curvature.
Frequency bias affects neural network training on non-uniform data.
problem Understanding how frequency bias impacts neural networks trained on non-uniformly distributed data.
method Used the Neural Tangent Kernel (NTK) model to explore the effect of variable density on training dynamics.
result Convergence time for learning a pure harmonic function depends on the local density at a point.
The paper provides a uniform convergence bound for smooth calibration error and its relationship with functional gradient.
problem Limited theoretical understanding of learning algorithms achieving high accuracy and good calibration.
method Focuses on smooth calibration error, providing a uniform convergence bound and proving the relationship with functional gradient.
result Derives conditions for simultaneous classification and calibration guarantees in gradient boosting trees, kernel boosting, and neural networks.