INT benchmark tests theorem proving agents' ability to generalize to unseen theorems.
problem Evaluating theorem proving agents' ability to generalize to unseen theorems.
method INT benchmark based on a theorem generation and proof procedure with adjustable knobs for measuring 6 types of generalization.
result MCTS can help agents prove new theorems.
Study noisy rewards in online decision-making with unknown distributions.
problem Learning optimal decisions in online settings with noisy and unknown reward distributions.
method Proposes algorithms integrating learning and decision-making via LCB thresholding.
result Achieves competitive ratios of 1 - 1/e and 1/2 in various settings.
Improved estimation of concentration using half-spaces for adversarial vulnerability.
problem Understanding the concentration of measure phenomenon and its impact on adversarial vulnerability.
method Extending Gaussian Isoperimetric Inequality to non-spherical Gaussian measures and arbitrary ℓ_p-norms, using half-spaces to estimate concentration.
result Proposed method finds tighter intrinsic robustness bounds, providing evidence against concentration as a cause of adversarial vulnerability.
Canary optimizes VaR-constrained RL problems with a conservative bound using Cantelli's inequality.
problem Optimizing reinforcement learning policies under VaR constraints in dense cost regimes.
method Employing Cantelli's inequality to create a conservative and smooth bound on VaR constraints based on moments of cost returns. Extending trust-region framework for worst-case bounds on policy improvement and constraint violation.
result Canary reliably satisfies VaR constraints with fewest violations and earliest permanent satisfaction, while maintaining reward competitiveness.
The prominent inequality of wealth and income is a huge concern especially in the United States. The likelihood of diminishing poverty is one valid reason to reduce the world's surging level of economic inequality. The principle of universal moral equality ensures sustainable development and improve the economic stabil…
ST-BCP narrows the coverage gap in BCP by transforming nonconformity scores.
problem The looseness in BCP's coverage guarantee due to Markov's inequality.
method Introduces a data-dependent transformation of nonconformity scores.
result Reduces the average coverage gap from 4.20% to 1.12% on benchmarks.
The data processing inequality doesn't always hold in practice, showing benefits in low-level tasks.
problem The data processing inequality suggests no benefit in pre-processing for classification.
method Theoretical and empirical study of binary classification setup with deep neural networks.
result Pre-classification processing can improve classification accuracy for any finite number of training samples.
New methods improve multi-agent reinforcement learning by addressing rotational dynamics.
problem Reproducibility crisis in multi-agent reinforcement learning.
method Reframing MARL approaches using Variational Inequalities (VIs) and proposing gradient-based VI methods.
result Significant performance improvements across benchmarks, including better convergence to equilibrium strategies in zero-sum games.
Concentration of infinitely exchangeable sequences with bounded-difference constants
problem Quantifying uncertainty in AI benchmarks
method Using a mixture-free Hoeffding-type bound
result Tight, mixture-free Hoeffding-type bound for zero-sum linear contrasts
Study optimal dividend and capital injection in insurance portfolios with self-exciting claim arrivals.
problem Optimal dividend and capital injection in insurance portfolios with Hawkes process claim arrivals.
method Analytical properties, explicit threshold, HJB variational inequality, finite-difference scheme, policy-gradient, actor-critic methods.
result Learned strategies closely match the PDE benchmark and remain stable across initial conditions.
The isoperimetric inequality and related inequalities are explored.
problem Proving the isoperimetric inequality and related inequalities.
method Discussing classical and recent proofs.
result Various proofs of the isoperimetric inequality and Sobolev inequality.
New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
Locally private methods detect changes in time series data.
problem Detecting distributional changes in time series data under local differential privacy.
method Proposed locally differentially private algorithms based on randomized response and binary mechanisms.
result Theoretical performance bounds and empirical validation of detection accuracy.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.
Unified framework for constrained diffusion models on nonconvex sets with efficient landing mechanism.
problem Efficiently modeling generative models under nonconvex constraints.
method Unified framework with overdamped and underdamped dynamics, landing mechanism.
result Significantly reduces computational cost while maintaining sample quality.
The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.
problem Proving (p,q)-Sobolev and Nash inequalities on Finsler metric measure manifolds. method Global p-Poincaré inequality, (p,q)-Sobolev inequality, Nash inequality derivation. result Established global optimal (p,q)-Sobolev inequality with a sharp constant. New inequality on sphere generalizes circle inequality.
problem Generalizing circle inequality to sphere.
method Develops a new inequality on the sphere that incorporates mass center deviation.
result Improves Aubin's inequality and Onofri's inequality.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Explains geometric inequalities for minimal hypersurfaces.
problem Geometric inequalities for minimal hypersurfaces.
method Expository discussion of known inequalities.
result Discussion of classical inequalities for minimal hypersurfaces.
Embed-KCPD segments text without labels, outperforming baselines.
problem Unsupervised text segmentation is challenging due to lack of boundary labels.
method Embed-KCPD represents sentences as embeddings and estimates boundaries via penalized KCPD.
result Embed-KCPD outperforms strong unsupervised baselines across benchmarks.
GenFlow optimizes faster, avoiding saddle points in fixed time.
problem Designing efficient optimization algorithms for convex and non-convex functions.
method Introduces GenFlow and momentum variants with fixed-time convergence guarantees.
result GenFlow and momentum variants converge to optimal solutions in fixed time for PL functions and evade saddle points uniformly.
The paper finds new inequalities for convex polygons.
problem Finding precise inequalities for convex polygons.
method Analytic isoperimetric inequalities based on Schur convex functions, followed by Bonnesen-style and inverse Bonnesen-style inequalities.
result Sharp discrete isoperimetric inequalities for planar convex polygons.
The study improves Bochner inequality on Finsler manifolds to derive important inequalities.
problem Improving Bochner inequality on Finsler manifolds to derive new inequalities.
method Using improved Bochner inequality and its integrated form, the study derives a sharp Poincaré-Lichnerowicz inequality, a new proof for logarithmic Sobolev inequality, and an estimate of geodesic ball volumes.
result Derivation of new inequalities and estimates on Finsler manifolds.
AdamL optimizes deep learning models by incorporating loss function information.
problem Adaptive optimizers can suffer from poor generalization due to nonuniform gradient scaling.
method AdamL is a novel adaptive optimizer that considers loss function information for better generalization.
result AdamL achieves faster convergence or lower objective function values compared to other optimizers.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
Meta-learning approaches have been proposed to tackle the few-shot learning problem.Typically, a meta-learner is trained on a variety of tasks in the hopes of being generalizable to new tasks. However, the generalizability on new tasks of a meta-learner could be fragile when it is over-trained on existing tasks during …
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Sharp inequalities for star bodies in 2D space.
problem Understanding star bodies in 2D space.
method Sharp inequalities for star bodies in R2. result New inequalities and proofs for star bodies.
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
Study on functional inequalities on simple edge spaces.
problem Whether classical functional inequalities hold in simple edge spaces.
method Analyzing Sobolev and Poincaré inequalities, proving optimality of Sobolev constant.
result Optimality result concerning the B-constant of the Sobolev inequality.
Proves inequalities on curved spaces with positive curvature.
problem Proving inequalities on manifolds with nonnegative Ricci curvature.
method Analyzes manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Proves Heisenberg-Pauli-Weyl, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities.
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
Paper refines Talagrand inequality on Euclidean spaces.
problem Improving Talagrand inequality for Euclidean spaces.
method Symmetrization and alternative proof methods.
result Several refined functional inequalities derived.
New inequalities for convex curves with multiple geometric factors.
problem Establishing inequalities for convex curves with multiple geometric factors.
method Parametric isoperimetric-type inequalities for closed convex curves with parameter conditions and equality conditions.
result Derived new inequalities and improved versions of existing inequalities.
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.
problem Functional and geometric inequalities on Finsler measure spaces.
method Local uniform Poincaré and Sobolev inequalities, mean value inequality, Harnack inequalities, and gradient estimates.
result Global gradient estimates for positive harmonic functions on Finsler measure spaces.
The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.
problem Proving a Harnack inequality for positive solutions to heat equations on Finsler metric measure manifolds.
method Volume comparison theorem, weighted Poincaré inequality, local uniform Sobolev inequality, mean value inequalities.
result Derives a Harnack inequality for positive solutions to heat equations.
Extends Riemannian geometry inequalities with sharper estimates.
problem Deriving new inequalities on Riemannian manifolds.
method Investigates advanced Hardy and Rellich-type inequalities on complete noncompact manifolds with weight functions.
result Provides sharper estimates conforming to the geometry and structure of the manifold.
The paper explores how information geometry impacts classical CR inequalities.
problem Deriving and generalizing CR inequalities using information geometry.
method Examining Eguchi's theory and applying Amari-Nagoaka's theory to KL-divergence, and then extending to other divergences.
result Generalized CR inequalities derived from various divergences.
The paper proves inequalities for hypersurfaces in weighted manifolds.
problem Willmore-type inequalities for closed hypersurfaces in weighted manifolds.
method Analyzes weighted manifolds with nonnegative Bakry-Émery Ricci curvature, proving sharp inequalities and characterizing equality cases.
result Derives sharp Willmore-type and Willmore-like inequalities in steady and shrinking gradient Ricci solitons.
Extends Gromov's optimal systolic inequality to manifolds with specific cohomology properties.
problem Finding optimal systolic inequalities for manifolds with complex cohomology structures.
method Extends Gromov's inequality to manifolds with fundamental cohomology classes as cup products of 2-dimensional classes.
result Provides an optimal systolic inequality for a new class of manifolds.
In this paper we presented a novel constructive approach for training deep neural networks using geometric approaches. We show that a topological covering can be used to define a class of distributed linear matrix inequalities, which in turn directly specify the shape and depth of a neural network architecture. The key…
Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.
problem Investigate functional and geometric inequalities on hyperbolic spaces and Riemannian manifolds.
method Employ symmetrization and semigroup approach based on sharp estimates for heat semigroup.
result Developed robust inequalities and methods relying on geometric and isoperimetric properties.
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
Sharp inequalities on curved spaces with bounded curvature.
problem Establishing inequalities on curved spaces with curvature constraints.
method Using Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below.
result Best constants for inequalities on curved spaces with curvature constraints.
Derives Weyl law for volume spectrum using parametric inequalities.
problem Deriving the Weyl law for the volume spectrum in compact Riemannian manifolds.
method Proves parametric generalizations of isoperimetric and coarea inequalities to derive the Weyl law.
result Derives the Weyl law for 1-cycles in 3-manifolds.
Extended a mathematical inequality by Andrews.
problem Mathematical inequality by Andrews.
method Proved extensions of Andrews inequality.
result Extended mathematical inequality by Andrews.