Proves inequalities for tensor fields on submanifolds using ABP method.
problem Proving Michael-Simon inequalities for tensor fields.
method Alexandrov-Bakelman-Pucci (ABP) method
result Proved Michael-Simon inequalities for tensor fields.
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48 results for “ABP method”
Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.
problem Proving logarithmic Sobolev inequalities on manifolds with nonnegative curvature.
method Employing the ABP method developed by Brendle.
result Sharp and logarithmic Sobolev inequalities established.
Sharp ABP estimate on metric spaces via optimal transport.
problem Sharp ABP estimate on metric measure spaces.
method Optimal transport theory.
result Established a sharp ABP estimate on metric measure spaces.
The ABP method is used to prove geometric inequalities for submanifolds and tensors.
problem Establishing geometric inequalities for submanifolds and tensors.
method Application of the Alexandrov-Bakelman-Pucci (ABP) method.
result Logarithmic Sobolev inequality and Sobolev-type inequality for submanifolds and tensors.
Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.
problem Developing an ABP approach to Sobolev and Michael-Simon inequalities under volume noncollapsing assumptions.
method Using a refinement of Brendle's contact-set argument to derive lower bounds for the volumes of geodesic balls.
result A Michael-Simon type inequality for immersed submanifolds with nonnegative sectional curvature and volume noncollapsing.
New Sobolev inequalities found for curved spaces.
problem Sobolev inequalities in curved spaces with specific decay conditions.
method Used ABP method developed by Cabré and Brendle.
result Established Sobolev inequalities for compact domains and submanifolds.
Deep Reinforcement Learning (RL) is proven powerful for decision making in simulated environments. However, training deep RL model is challenging in real world applications such as production-scale health-care or recommender systems because of the expensiveness of interaction and limitation of budget at deployment. One…
Employing a notion of curvature for arbitrary closed sets we prove an ABP-type estimate for a class of singular submanifolds of arbitrary codimension and bounded mean curvature recently introduced by B. White. A weak-Harnack-type estimate is then derived using the ABP estimate. These results generalize analogous result…
Sharp inequalities for manifolds with nonnegative curvature.
problem Establishing inequalities for manifolds with nonnegative curvature.
method Using the ABP-method, generalizing previous work by Brendle.
result Sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds.
Rare dense solutions clusters in asymmetric binary perceptrons -- local entropy via fully lifted RDTstat.ML
New method finds rare dense clusters in asymmetric binary perceptrons, resolving algorithmic hardness.
problem Resolving algorithmic hardness in asymmetric binary perceptrons.
method Fully lifted random duality theory (fl RDT) and large deviation upgrade (sfl LD RDT).
result Local entropy breaks down for constraint densities in (0.77, 0.78) interval, matching current solver limits.
New estimate for complex Monge-Ampère equations improves previous results.
problem Improving estimates for complex Monge-Ampère equations.
method Using the ABP maximum principle to prove a new gradient estimate.
result Proves a new gradient estimate for complex Monge-Ampère equations.
Machine learning predicts phase behavior in active matter suspensions.
problem Predicting phase behavior in active matter systems using machine learning.
method Used deep learning techniques, including fully connected networks and graph neural networks, to predict motility-induced phase separation (MIPS) in ABP suspensions.
result Strong agreement between machine learning predictions and MIPS binodal from simulations, suggesting machine learning as an effective method for phase behavior determination.
Gradient estimate proved for Donaldson's equation on Kähler manifolds.
problem Proving gradient estimates for Donaldson's equation on compact Kähler manifolds.
method Using uniform upper bounds for and Alexandrov-Bakelman-Pucci (ABP) maximum principle.
result Gradient estimate for Donaldson's equation derived from uniform bounds.
Sharp inequality outside ball proved using Neumann method.
problem Anisotropic isoperimetric inequality for domains outside an Euclidean ball.
method Applied ABP method to Neumann boundary value problem.
result Proved sharp anisotropic isoperimetric inequality.
The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.
problem Analyzing geometric properties of weighted manifolds under Ricci curvature bounds.
method Develops geometric analysis techniques on weighted Riemannian manifolds with lower -weighted Ricci curvature bounds.
result Proves eigenvalue estimates for Steklov and ABP inequalities on weighted manifolds.
We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of . Our result applies to…
Sobolev inequalities involving 2-tensor fields in manifolds with nonnegative sectional curvaturemath.DG
Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.
problem Establishing inequalities for tensor fields on curved manifolds.
method Applying the ABP method to symmetric tensor fields on manifolds with nonnegative sectional curvature.
result Log Sobolev and Michael Simon inequalities for tensor fields.
New insights into binary perceptron reveal phase transitions and algorithmic thresholds.
problem Understanding the statistical-computational gap in binary perceptron models.
method Application of fully lifted random duality theory (fl RDT) to uncover structural changes.
result Numerical estimates of constraint density thresholds align with theoretical predictions.
Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvaturemath.DG
The study proves inequalities on curved spaces without global curvature bounds.
problem Proving inequalities on manifolds with non-negative curvature outside compact sets.
method ABP method localized to regions of non-negative curvature, spectral properties of manifolds.
result Validated isoperimetric and Michael-Simon inequalities on manifolds with asymptotically non-negative curvature.
New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.
problem Finding a Sobolev inequality for mean convex spacelike submanifolds in Minkowski space.
method Applying the ABP estimate method to spacelike submanifolds in .
result Obtained a Sobolev inequality without a mean curvature term for mean convex hypersurfaces.
Proves inequality for tensor fields on curved spaces.
problem Generalizing inequality for tensor fields on curved spaces.
method Alexandrov-Bakelman-Pucci (ABP) method
result Proves Michael-Simon-Sobolev inequality for tensor fields.
New proofs and inequalities for capillarity problems quantify asymmetries.
problem Quantifying asymmetries in capillarity functionals.
method ABP-type technique, symmetrization, selection-type argument.
result Sharp quantitative inequalities for asymmetries in capillarity problems.
Asymptotically non-negative Ricci curvature, elliptic Kato constant and isoperimetric inequalitiesmath.DG
The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.
problem Proving isoperimetric inequalities in manifolds with small negative Ricci curvature.
method Expanding on the ABP method, the paper uses the elliptic Kato constant to control the non-negativity of the Ricci-tensor and applies techniques from Li-Tam and Kasue.
result Sharp isoperimetric inequalities in the limit are proven in the presence of small negative curvature.
New algorithm learns optimal policy with multi-step lookahead information.
problem Learning optimal policy in reinforcement learning with multi-step lookahead information is NP-hard.
method Adaptive batching policies that process lookahead in state-dependent chunks.
result Order-optimal regret bounds up to a constant factor of lookahead horizon.
Estimates complex Hessian integral for complex Monge-Ampère equations.
problem Improving classical ABP estimate for complex settings.
method De Giorgi iteration method for complex Monge-Ampère equations.
result Sharp gradient estimates for complex Monge-Ampère equations.
On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-Émery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-lin…
Logarithmic Sobolev inequality proven for non-compact self-shrinkers.
problem Establishing a logarithmic Sobolev inequality for non-compact self-shrinkers.
method Using Alexandrov-Bakelman-Pucci (ABP) method to prove the inequality for Euclidean space, then applying this method to non-compact self-shrinkers.
result Optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers.
Gradient bounds and Liouville theorems for quasi-linear equations on manifolds with nonnegative Ricci curvature.
problem Establishing bounds and theorems for solutions to quasi-linear elliptic equations on compact manifolds with nonnegative Ricci curvature.
method Gradient bounds, Liouville-type theorems, local splitting theorem, Harnack-type inequality, ABP estimate.
result Gradient bounds and Liouville-type theorems for solutions to quasi-linear equations on compact manifolds with nonnegative Ricci curvature.
In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds with free boundary. We first generalize ideas of restricted normal cones introduced by Choe-Ghomi-Ritoré in \cite{CGR06} and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, tog…
In this article we construct a family of genus two Lefschetz fibrations with , , and by applying a single lantern substitution to the twisted fiber sums of Matsumoto's genus two Lefschetz fibration over .…
Study potential computational gaps in symmetric binary perceptrons using fl-RDT.
problem Potential statistical-computational gaps in symmetric binary perceptrons.
method Parametric utilization of fully lifted random duality theory (fl-RDT).
result Observation of a computational gap in SBP.
Improved iterative methods for risk parity portfolio weights.
problem Solving for portfolio weights in risk parity allocation.
method Enhanced CCD and Newton methods, including a rescaling step and improved initial guess.
result Improved CCD method is the best, three times faster with 40% fewer iterations.
We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems. Our method has only one parameter (a step size), and is radically simpler than o…
A new method combines Laplace and Variational Bayes for scalable inference.
problem Complex models and large datasets make exact inference infeasible.
method Low-Rank Variational Bayes Correction (VBC) using Laplace method and Variational Bayes correction in a lower dimension.
result The method ensures scalability in both model complexity and data size.
Unified framework for model explanation methods based on feature removal.
problem Unclear relationships and preferences among various model explanation methods.
method Characterizes removal-based explanations along three dimensions.
result Unified 26 existing methods, including widely used approaches.
This work reviews and evaluates methods for predicting prediction intervals in regression problems.
problem Calibration of prediction intervals in regression problems.
method Four classes of methods: Bayesian, ensemble, direct interval estimation, and conformal prediction.
result Conformal prediction can be used as a general calibration procedure.
Derives kernel PCA with Nyström method for scalability.
problem Scalability of kernel PCA.
method Nyström method for kernel PCA.
result Provides scalable alternative to full kernel PCA.
In this paper, the author considers the numerical computation of CVA for large systems by Mote Carlo methods. He introduces two types of stochastic mesh methods for the computations of CVA. In the first method, stochastic mesh method is used to obtain the future value of the derivative contracts. In the second method, …
New method combines spectral and sparse methods for Gaussian processes.
problem Efficiently fitting Gaussian processes to large datasets.
method Orthogonally decoupled variational Fourier features.
result Competitive performance on synthetic and real-world data.
A Large-Scale Comparative Analysis of Imputation Methods for Single-Cell RNA Sequencing Dataq-bio.GN
A comprehensive benchmark of 15 scRNA-seq imputation methods across various datasets and analyses.
problem Imputation of single-cell RNA sequencing data to recover latent transcriptional signals.
method Evaluation of 15 imputation methods across 30 datasets and 6 downstream analyses.
result Traditional methods generally outperform DL-based methods in scRNA-seq data analysis.
New methods using natural gradient for structured optimization.
problem Structured optimization problems.
method Structured second-order methods via natural gradient descent.
result Efficiency demonstrated on non-convex and deep learning problems.
Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimizationmath.OC
Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of t…
We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference me…
A new method speeds up deep neural network training.
problem Nonconvex optimization in deep neural networks.
method Scaled conjugate gradient method for nonconvex optimization.
result The method converges faster and achieves lower scores in practical applications.
We propose a new stochastic dual coordinate ascent technique that can be applied to a wide range of regularized learning problems. Our method is based on Alternating Direction Multiplier Method (ADMM) to deal with complex regularization functions such as structured regularizations. Although the original ADMM is a batch…
NCG methods improve shape optimization efficiency.
problem Shape optimization problems
method Nonlinear conjugate gradient methods
result NCG methods are efficient for shape optimization
Pricing Derivatives under Multiple Stochastic Factors by Localized Radial Basis Function Methodsq-fin.CP
We propose two localized Radial Basis Function (RBF) methods, the Radial Basis Function Partition of Unity method (RBF-PUM) and the Radial Basis Function generated Finite Differences method (RBF-FD), for solving financial derivative pricing problems arising from market models with multiple stochastic factors. We demons…
Uncertainty Aware Trader-Company Method: Interpretable Stock Price Prediction Capturing Uncertaintyq-fin.CP
Proposes UTC method for stock price prediction with uncertainty quantification.
problem Lack of uncertainty estimates in stock prediction methods.
method Combines TC method with probabilistic modeling for point and uncertainty predictions.
result UTC method achieves higher returns and lower risks than baselines.