New PAC-Bayesian bounds provide practical guarantees for neural networks.
problem Loose derandomization step in PAC-Bayesian bounds for deterministic models.
method Introduce disintegrated PAC-Bayesian bounds for deterministic models.
result Significant practical improvement over state-of-the-art bounds.
New tools for constructing disintegrations and studying their modes.
problem Difficulty in constructing disintegrations and understanding their modes.
method Developed comprehensive mathematical tools for constructing disintegrations and analyzing their modes.
result Disagreement between restricted density and disintegration density in certain cases.
Proves uniqueness of barycenters on manifolds without restrictions.
problem Finding unique barycenters on complex geometric spaces.
method Introduces new disintegrated Monge-Kantorovich metrics for barycenter problems.
result Uniqueness of barycenters on connected, complete Riemannian manifolds.
Improved bounds on learning algorithms' performance using conditional mutual information.
problem Bounding the generalization error of learning algorithms.
method Introducing conditional mutual information and disintegrated mutual information to tighten bounds.
result New bounds are tighter than previous ones, especially for noisy, iterative algorithms.
Classifies invariant measures on specific character varieties.
problem Classifying invariant probability measures on character varieties.
method Measure disintegration along transverse Lagrangian tori fibrations.
result Ergodic measures are either counting measures on finite orbits or Liouville measures.
New bounds improve neural network generalization through slicing.
problem Difficulty in evaluating mutual information in high dimensions for neural networks.
method Slicing the parameter space and using disintegrated mutual information and k-sliced mutual information.
result Slicing improves generalization and offers significant computational and statistical advantages.
In this paper we study arbitrage theory of financial markets in the absence of a numéraire both in discrete and continuous time. In our main results, we provide a generalization of the classical equivalence between no unbounded profits with bounded risk (NUPBR) and the existence of a supermartingale deflator. To obtain…
The study proves curvature bounds for quotient spaces of isometric actions.
problem Proving curvature bounds for quotient spaces of isometric actions.
method Disintegrate absolutely continuous measures and define a functional to prove curvature bounds.
result Necessary and sufficient conditions for Ricci curvature to be bounded below.
New bounds for model generalization under deterministic gradient descent.
problem Establishing generalization bounds for models trained with gradient descent methods.
method PAC-Bayesian bounds for deterministic optimisation algorithms.
result Fully computable bounds that depend on initial distribution and Hessian.
Paper develops a new generalization bound using PAC-Bayes theory and Gibbs distributions.
problem Limits of traditional generalization bounds due to complexity measures.
method Leverages PAC-Bayes bounds with Gibbs distributions to derive a flexible generalization bound.
result Derives a generalization bound that can adapt to both hypothesis class and task complexity.
Study heat content on RCD(K,N) spaces with specific boundary conditions.
problem Analyzing heat content in RCD(K,N) spaces with irregular boundaries.
method Proved first-order asymptotics using measured interior geodesic condition.
result Established first-order heat content asymptotics on RCD(K,N) spaces.
New risk measures control subgroup imbalances, improving PAC-Bayesian bounds.
problem Insufficient risk bounds for subgroup imbalances in data.
method Introduce constrained f-entropic risk measures and derive PAC-Bayesian bounds.
result First disintegrated PAC-Bayesian guarantees beyond standard risks.
Oral Disintegrating Tablets (ODTs) is a novel dosage form that can be dissolved on the tongue within 3min or less especially for geriatric and pediatric patients. Current ODT formulation studies usually rely on the personal experience of pharmaceutical experts and trial-and-error in the laboratory, which is inefficient…
The paper extends localisation technique to multiple constraints in Euclidean spaces.
problem Proving log-concavity of conditional measures in decomposed convex sets.
method Defining partitions of maximal closed convex sets and proving log-concavity of conditional measures.
result Existence of a partition and log-concavity of conditional measures for almost every set of the partition.
New bounds using samplewise evaluated CMI for deep neural networks.
problem Improving generalization bounds for deep neural networks.
method Introduced a new family of information-theoretic generalization bounds using samplewise evaluated conditional mutual information (CMI).
result The new bounds can be tighter than previous ones for deep neural networks.
We consider the question of learning in general topological vector spaces. By exploiting known (or parametrized) covariance structures, our Main Theorem demonstrates that any continuous linear map corresponds to a certain isomorphism of embedded Hilbert spaces. By inverting this isomorphism and extending continuously, …
The curve graph and related graphs are hyperbolic and have quasi-tree fibers.
problem Understanding the structure of the curve graph and related graphs.
method Analyzing a sequence of graphs with Lipschitz maps and proving hyperbolicity and quasi-tree properties.
result The graphs in the sequence are hyperbolic and have quasi-tree fibers, leading to bounds on asymptotic dimension and acylindrical actions.
Given a pair of second order diffusion operators, one on the total space of a principle bundle N and the other on the base space M, intertwined by the projection π:N→M, if the operator A on the base manifold has constant rank, we define a semi-connection on the principal bundle which allows to spl…
Paper addresses the disparity between sampled and mean representations in disentangled learning.
problem Disparity between sampled and mean representations in disentangled learning.
method Proposes a method to eliminate the disparity by proving and utilizing the relationship between total correlation of sampled and mean representations for multivariate normal distributions.
result Demonstrates that a factorized mean representation can have lower total correlation than the sampled representation.
Study SRB measures for Anosov actions on manifolds.
problem Characterize SRB measures for Anosov actions.
method Use Ruelle-Taylor resonances and properties of Sinai-Ruelle-Bowen measures.
result SRB measures have properties like smooth disintegrations, positive basins, and are unique under certain conditions.
For a given 1-Lipschitz map u:Rn→Rm we define a partition, up to a set of Lebesgue measure zero, of Rn into maximal closed convex sets such that restriction of u is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave meas…
Trade finance history traced from medieval origins to modern markets.
problem Evolution and standardization of trade finance products.
method Historical analysis of market structures and regulatory changes.
result Global trade finance market evolved from local to centralized, then decentralized.
Develops a new framework for conditional independence.
problem Generalizing previous notions of conditional independence.
method Introduces transition probability spaces and transitional random variables.
result Satisfies all desired relevance relations except symmetry.
Study approximates operators on labelled conditional distributions for non-exchangeable systems.
problem Approximating operators on constrained probability measures for non-exchangeable systems.
method Combines cylindrical approximations and DeepONet-type neural architecture for finite-dimensional representations.
result Establishes a universal approximation theorem for continuous operators on Mλ. New method reduces memory usage for Bayesian inverse problems on large grids.
problem Solving large-scale linear inverse problems with Gaussian process priors.
method Implicit representation of posterior covariance matrices, sequential disintegrations of Gaussian measures.
result Significant reduction in uncertainty for high-density regions estimation.
The paper analyzes financial market turbulence using mathematical physics.
problem Understanding price fluctuations caused by information asymmetry.
method Spectrum analysis to decompose pricing patterns.
result Identifies phase correlations in financial stock market turbulence.
The study optimizes bounds for comparing training and population loss.
problem Optimizing bounds for comparing training and population loss.
method Derives generic information-theoretic and PAC-Bayesian generalization bounds using convex comparator functions.
result The tightest possible bound is obtained with the comparator being the convex conjugate of the CGF of the bounding distribution.
Introduces bounded scale measure and generalizes property A.
problem Defining property A for large scale spaces with bounded geometry.
method Introduces bounded scale measure, shows its coarse invariance, and generalizes property A.
result Definition of property A for large scale spaces with bounded scale measure is a coarse invariant.
Paper improves PAC-Bayes bounds for various loss types.
problem Improving PAC-Bayes bounds for different types of losses.
method Introducing new high-probability PAC-Bayes bounds for bounded and general tail behaviors losses, and extending to anytime-valid bounds.
result New fast-rate and mixed-rate bounds for losses with bounded ranges, and parameter-free bounds for losses with general tail behaviors.
Improved bounds for Monte Carlo Rademacher Averages using self-bounding functions.
problem Proving sharper concentration bounds for MCERA.
method Deriving new bounds through self-bounding functions and concentration of measure.
result Novel bounds depend on data-dependent quantities, improving over standard methods.
Study bounds on self-shrinkers with bounded HA for applications.
problem Understanding bounds on self-shrinkers with bounded HA.
method Integral and pointwise bounds on the second fundamental form of self-shrinkers.
result Gap and compactness results for self-shrinkers.
Investigates tight PAC-Bayes bounds for small datasets.
problem Tightening PAC-Bayes bounds for small data.
method Generic PAC-Bayes theorem, meta-learning, synthetic tasks.
result PAC-Bayes bounds are competitive with Chernoff bounds but not as tight.
Extends Fatou theorem to bounded harmonic maps.
problem Classical Fatou theorem for bounded harmonic functions.
method Extending theorem to bounded harmonic maps.
result Identifies bounded harmonic maps on unit disk with bounded measurable functions on boundary.
New bound relaxes uniform gradient norm assumptions for PAC-Bayesian bounds.
problem Generalization bounds with strict assumptions like uniformly bounded loss.
method Relax uniform bounds assumptions to on-average bounded loss and gradient norm.
result Proposes a new generalization bound with a surrogate of model complexity.
Jiang et al. (2020) found no uniformly tight generalization bounds for neural networks in the overparameterized setting.
problem Finding uniformly tight generalization bounds for neural networks in the overparameterized setting.
method Examined more than a dozen generalization bounds, proving that no bounds can be uniformly tight in the overparameterized setting.
result No generalization bounds can be uniformly tight in the overparameterized setting.
Willmore-type inequalities for bounded domains in manifolds with curvature bounds.
problem Establishing inequalities for bounded domains in manifolds with curvature bounds.
method Using asymptotic or integral Ricci curvature bounds to establish inequalities.
result Recovering a recent inequality of Jin-Yin.
Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
Paper improves SLCB regret bound for bounded noise.
problem Stochastic linear contextual bandits with bounded noise.
method Set-membership estimation (SME) and optimism in the face of uncertainty (OFU).
result Improved regret bound of O(logT). Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.
problem Understanding CMC hypersurfaces with bounded index and area.
method Bubble-compactness theory for embedded CMC hypersurfaces in low dimensions.
result Minimal blow-ups are all catenoids, and bounds on genus provided.
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.
New study on regret lower bounds for multi-agent multi-armed bandit problems.
problem Understanding the limits of performance in multi-agent multi-armed bandit problems.
method Comprehensive study on different settings, establishing tight lower bounds.
result First comprehensive study on regret lower bounds across various settings.
The paper improves PAC-Bayes bounds for losses with finite moments.
problem Bounding generalization for losses with heavy tails and finite moments.
method Truncation method and PAC-Bayes bounds for unbounded losses with heavy tails and bounded variance.
result Bounds interpolate between slow and fast rates depending on the moment.
Sharp lower bound for Hodge Laplacian on Kähler hyperbolic manifolds.
problem Finding a sharp lower bound for the spectrum of the Hodge Laplacian.
method Explicitly expressed in terms of the supremum norm of the 1-form.
result Explicit spectral lower bounds for bounded symmetric domains.
New bounds for SGD show improved performance in various settings.
problem Improving convergence bounds for SGD with random permutations.
method Analyzing convergence of SGD with random reshuffling and arbitrary permutations.
result Tighter lower bounds for weighted average iterates in both convex and strongly-convex cases.
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.
The paper honors Lai's contributions to multi-armed bandits and establishes new regret bounds.
problem Improving regret bounds in multi-armed bandit problems.
method Establishes non-asymptotic regret bounds for upper confidence bound indices.
result New regret bounds match Lai-Robbins lower bound.
New method to parametrize infinite Riemann surfaces with bounded triangulations.
problem Parametrizing infinite Riemann surfaces with bounded triangulations.
method Introducing bounded ideal triangulations and proving real-analyticity of the parametrization.
result Real-analytic parametrization of Teichmüller spaces for infinite surfaces with bounded triangulations.
We propose a general framework for studying adaptive regret bounds in the online learning framework, including model selection bounds and data-dependent bounds. Given a data- or model-dependent bound we ask, "Does there exist some algorithm achieving this bound?" We show that modifications to recently introduced sequen…