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.
We revisit Spakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology. We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with…
Uniform entropy bound for Ricci shrinkers with bounded curvature.
problem Bounding entropy for Ricci shrinkers with specific curvature constraints.
method Establishing uniform entropy bounds for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
result Uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
Uniform bounds derived for fully non-linear equations.
problem Bounding fully non-linear equations uniformly in background metrics.
method Auxiliary Monge-Ampère equations and entropy-like quantities.
result Uniform L∞ bounds for systems coupling fully non-linear equations to their linearizations. Logistic regression gets a new, simpler uniform bound.
problem Finding a uniform bound for logistic regression's empirical risk.
method PAC-Bayes approach with second-order expansion and Rademacher-complexity bounds.
result Provides a dimension-free uniform concentration bound.
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.
This work studies the robustness certification problem of neural network models, which aims to find certified adversary-free regions as large as possible around data points. In contrast to the existing approaches that seek regions bounded uniformly along all input features, we consider non-uniform bounds and use it to …
A new method achieves optimal uniformity in designs with minimal flexibility.
problem Achieving optimal uniformity in designs with minimal flexibility.
method Derive a lower bound on the uniformity constant and use a greedy construction to achieve this bound, then extend the scheme for more flexibility.
result A simple greedy construction achieves the optimal uniformity constant.
Uniformizes klt pairs using bounded symmetric domains.
problem Characterizing klt pairs uniformizable by bounded symmetric domains.
method Determines conditions for uniformization using Miyaoka-Yau-type inequalities.
result Characterizations of orbifold quotients of polydisc and classical bounded symmetric domains.
We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. This generalization will follow as a corollary from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the un…
Uniform bounds for complex equations using Monge-Ampère method.
problem Bounding solutions to complex equations.
method Auxiliary Monge-Ampère equation method.
result Uniform bounds remain valid even as background metrics degenerate.
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.
We prove a uniform extension result for contracting maps defined on subsets of Hadamard manifolds subject to curvature bounds.
Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.
problem Bounding the diameter of Calabi-Yau fibrations near singular fibers.
method Uniform diameter bound proof for Calabi-Yau fibrations with canonical singular fibers.
result Uniform diameter bounds for all fibres in suitable rescaling.
New uniformity definition on noncompact manifolds without metrics.
problem No Riemannian metric for uniformity on noncompact manifolds.
method Definition of uniformity without metrics, equivalent to bounded geometry.
result Equivalent definition of uniformity on noncompact manifolds.
We prove that if Y is the Gromov-Hausdorff limit of a sequence of compact manifolds, Min, with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then Y has a universal cover. We then show that, for i sufficiently large, the fundamental group of Mi has a surjective homeomorphis…
New algorithms achieve uniform-PAC guarantees for RL with bounded eluder dimension.
problem Achieving strong performance guarantees in reinforcement learning.
method Proposes algorithms for nonlinear bandits and model-based episodic RL with a bounded eluder dimension.
result Achieves uniform-PAC sample complexity that matches state-of-the-art regret bounds or sample complexity guarantees.
The study examines metrics on Riemannian spaces with bounded properties and finds conditions for Lipschitz and uniform bounds.
problem Investigating bounded rough Riemannian metrics and their properties.
method Analyzing the structure of bounded rough Riemannian metrics and finding conditions for Lipschitz and uniform bounds.
result Weak conditions are identified for Lipschitz and uniform bounds on the metrics.
Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.
problem Establishing uniform bounds for eigenvalues of the Hodge Laplacian on manifolds with specific geometric constraints.
method Using uniform bounds on Ricci curvature, injectivity radius, and diameter, we derive eigenvalue estimates for the Hodge Laplacian.
result Uniform upper bounds for eigenvalues of the Hodge Laplacian on differential forms on manifolds with given geometric constraints.
PAC-Bayesian theory applied to data-dependent hypothesis sets yields uniform generalization bounds.
problem Proving uniform generalization bounds for data-dependent hypothesis sets.
method Applying PAC-Bayesian framework on 'random sets' and considering data-dependent hypothesis sets.
result Data-dependent uniform generalization bounds are proven, providing tighter and unified results.
Uniform diameter bound for reflection group disk patterns.
problem Uniform bounded diameter conjecture for reflection groups.
method Skinning map analysis and discrete extremal width on Coxeter graph.
result Diameter of skinning image is bounded by a constant.
A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over…
We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more general than similar classes defined by other authors. We revisit Spakula's uniform K-…
This paper refines homotopy theory for cubical sets and uniform spaces.
problem Classical homotopy theory limitations in cubical sets and uniform spaces.
method Develops a uniform-theoretic refinement for cubical sets and uniform spaces, lifting to a full and faithful embedding.
result Lifts classical homotopy categories to new uniform homotopy categories, generalizing cohomology theories.
Uniform linear bounds on volume changes in 3D hyperbolic spaces.
problem Volume variation in hyperbolic 3-manifolds.
method Uniform linear bounds proof for drilling and filling operations.
result Uniform linear bounds on volume variation proved.
Equivalence proven between uniformizing varieties and tensors, generalizing uniformization results.
problem Characterizing complex-projective varieties with klt singularities and ample canonical divisors.
method Constructing a uniformizing variation of Hodge structure from slope zero tensors and vice versa.
result Generalization of uniformization results to singular settings, including quotients of tube domains.
In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X, then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil-Petersson norm of μ over the square root of the systole…
Unified framework for anytime-valid PAC-Bayes bounds.
problem Deriving time-uniform PAC-Bayes bounds for stochastic processes.
method Combines four tools: nonnegative supermartingales, method of mixtures, Donsker-Varadhan formula, and Ville's inequality.
result Unified PAC-Bayes theorem for a wide class of discrete stochastic processes.
Uniform bounds on ends for non-branching CD spaces with nonnegative curvature outside a compact set.
problem Bounding the number of ends of non-branching CD spaces with nonnegative curvature outside a compact set.
method Adapting Z.-D. Liu's work to prove a ball covering property.
result Uniform bounds on the number of ends of such spaces.
New approach to adversarial robustness with non-uniform perturbations.
problem Real-world adversaries craft adversarial examples with non-uniform perturbations.
method Proposes non-uniform perturbations based on feature dependencies and data distribution.
result Shows improved robustness to real-world attacks compared to uniform perturbations.
We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our result holds for any metric with a uniform bound on the W2,p-harmonic radius…
Uniform bounds for eigenvalues of Hodge Laplacian on manifolds with lower Ricci curvature.
problem Establishing bounds for eigenvalues of Hodge Laplacian under lower Ricci curvature.
method Using geometric assumptions including lower Ricci curvature, injectivity radius, and diameter bounds.
result Uniform eigenvalue bounds for the Hodge Laplacian and connection Laplacian.
Develops Lefschetz theory for noncompact manifolds.
problem Lefschetz fixed-point theory for noncompact manifolds.
method Introduces uniform bounded cohomology and develops obstruction theory.
result Uniform Lefschetz class vanishes if and only if map is homotopic to a strongly fixed-point free map.
Paper extends learning theory to dependent data with uniform risk bounds.
problem Learning with dependent data sequences.
method Derives uniform risk bounds for dependent data using VC-dimension and Rademacher complexity.
result Standard classification risk bounds hold for dependent data, same as for independent data.
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.
The study explores maps of 2- and 3-uniform tilings on the torus.
problem Understanding the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
method Analyzing the quotient maps of 2- and 3-uniform tilings on the torus.
result Bounds on the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.
problem Bounding Nash entropy in ancient Ricci flows.
method Uniformly bounded Nash entropy implies uniform bounds on the ν-functional, leading to uniform logarithmic and Sobolev inequalities.
result Uniform logarithmic and Sobolev inequalities on ancient Ricci flows with bounded Nash entropy.
Uniform bounds for Green's function on Kähler manifolds derived from complex Monge-Ampère equations.
problem Uniform bounds for Green's function on Kähler manifolds.
method Auxiliary Monge-Ampère equations, non-linear proof.
result Uniform lower bounds for the Green's function on Kähler manifolds.
Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
New bounds on self-normalized martingales improve online linear regression performance.
problem Improving regret bounds in online linear regression.
method Characterizing scale-invariant bounds on self-normalized martingales.
result For d=1, O(logT) doubly-uniform regret is possible; for d>1, sublinear doubly-uniform regret is impossible. The paper derives uniform stability-based coverage bounds for conformal prediction methods.
problem Establishing theoretical guarantees for conformal prediction methods.
method Uniform stability perspective applied to full-conformal, jackknife+, and CV+ prediction regions.
result Coverage bounds for finite-dimensional models derived using a concentration argument.
Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.
problem Understanding singularity types in long-time solutions of Chern-Ricci flow.
method Extended results from Kähler-Ricci flow to Chern-Ricci flow, focusing on uniform bounds on torsion and curvature.
result Uniform bounds on torsion and curvature for solutions starting from metrics of the same ∂∂ˉ class. Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any suc…
We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K-homology, and that this class only depends on the principal symbol of the operator.
The paper achieves nearly optimal regret bounds for contextual multinomial logit bandits.
problem The contextual multinomial logit (MNL) bandit problem with varying rewards.
method Established lower bounds and proposed OFU-MNL+ algorithm with matching upper bounds.
result Achieved minimax optimal regret bounds for both uniform and non-uniform reward settings.
Study infinite genus surfaces and Schottky groups for uniformization.
problem Investigate infinite genus surfaces and Schottky groups for uniformization.
method Definitions and proofs for infinite genus surfaces and Schottky groups, showing uniformization by Schottky groups.
result Infinite genus surfaces and handlebodies can be topologically and quasiconformally uniformized by Schottky groups.
Uniform bounds found for Sierpinski carpet hyperbolic components.
problem Bounding hyperbolic components of Sierpinski carpet type.
method Establishing uniform a priori bounds and analyzing quadratic-like restrictions.
result Sierpinski carpet hyperbolic components of disjoint type are bounded.
Classifies 2-uniform maps on torus with formulas and asymptotic bounds.
problem Classifying 2-uniform maps on torus.
method Classification through arithmetic functions and asymptotic analysis.
result Explicit formulas and asymptotic bounds for 2-uniform maps on torus.