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.
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.
Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.
problem Volume noncollapsing for Kähler metrics induced by complex Monge-Ampère equations.
method Proves local volume noncollapsing estimate with Ricci curvature lower bound.
result Establishes diameter and gradient estimates for Kähler metrics.
Paper establishes L∞ estimates for complex Monge-Ampere and Hessian equations.
problem Estimating solutions to complex Monge-Ampere and Hessian equations.
method Uses PDE techniques similar to Phong et al to prove L∞ and Hölder estimates. result Establishes L∞ estimates for both complex Monge-Ampere and Hessian equations. Gradient and Laplacian estimates for complex Monge-Ampère equations found.
problem Estimating solutions to complex Monge-Ampère equations with singularities.
method Integral method applied to obtain gradient and Laplacian estimates.
result Gradient and Laplacian estimates for the solution to the singular complex Monge-Ampère equation.
Unified estimate for complex Monge-Ampère equations on Kähler manifolds.
problem Estimating solutions to complex Monge-Ampère equations on Kähler manifolds.
method Unified approach using PDE methods and entropy bounds to construct comparison metrics.
result Improves previous results on modulus of continuity, stability, and W1,1-estimates of Green's functions. 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.
New approach for uniform estimates in complex equations.
problem Uniform estimates for solutions to complex Monge-Ampere equations.
method Efficient new approach to uniform estimates.
result Efficient method for uniform estimates in geometric PDEs.
Note on gradient estimates for complex Monge-Ampere equation.
problem Gradient estimates for solutions of complex Monge-Ampere equation.
method Estimates Lp and L∞ for gradient in terms of continuity of the right-hand side. result Gradient estimates for solutions of complex Monge-Ampere equation.
New proof for stability estimates in complex equations without pluripotential theory.
problem Stability estimates for complex Monge-Ampère and Hessian equations.
method New proof using general degenerations of background metrics.
result Uniform stability estimates for both equations under various degenerations.
Sharp L∞ estimates proved for complex Monge-Ampère equations.
problem Proving sharp L∞ estimates for complex Monge-Ampère equations. method PDE proof covering fixed and degenerating background metrics, extends to general fully non-linear equations.
result Sharp L∞ estimates proved for complex Monge-Ampère equations. We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein met…
The paper proves strong holomorphic Morse inequalities on complex manifolds with optimal estimates.
problem Holomorphic Morse inequalities on non-compact complex manifolds with optimal fundamental estimates.
method Established strong holomorphic Morse inequalities under optimal fundamental estimates.
result Strong holomorphic Morse inequalities hold true on non-compact complex manifolds with optimal fundamental estimates.
Estimates covariance matrices using Markov chain Monte Carlo with improved sample complexity.
problem Complexity of covariance matrix estimation for Gibbs distributions.
method Uses Markov chain Monte Carlo with conditions on the chain's spectral gap and Poincaré inequality.
result Achieves similar sample complexity as i.i.d. samples with better query complexity.
C-SURE improves complex-valued deep learning models by shrinking estimates, outperforming MLE and SurReal.
problem Improving accuracy and robustness of complex-valued deep learning models.
method Proposes a Stein's unbiased risk estimate (SURE) for complex-valued data and integrates it into a prototype CNN classifier.
result C-SURE outperforms SurReal and MLE in accuracy and robustness on complex-valued datasets.
The note provides uniform estimates for complex Hessian equations on compact Hermitian manifolds.
problem Uniform estimates for solutions to degenerate complex Hessian equations on compact Hermitian manifolds.
method The approach relies on corresponding a priori estimates for Monge-Ampère equations.
result Extension and short alternative proof of results for complex Hessian equations.
Paper derives estimates for complex Hessian equations on Hermitian manifolds.
problem Estimating solutions to complex Hessian equations on Hermitian manifolds.
method Derives second order estimates for solutions in a specific cone.
result Establishes second order estimates for solutions in Γk+1 cone. Estimates neural network errors for classification problems.
problem Binary and multi-class classification problems.
method Rademacher complexity estimates and direct approximation theorems.
result A priori error estimates for regularized loss functionals.
ULA estimates covariance of log-concave distributions efficiently.
problem Estimating covariance matrices of log-concave distributions efficiently.
method Unadjusted Langevin algorithm (ULA) for sampling and covariance estimation.
result Sample complexity of single-chain ULA is smaller than that of parallel ULA by a logarithmic factor.
In this paper, we obtain the Bedford-Taylor interior C2 estimate and local Calabi C3 estimate for the solutions to complex Monge-Ampère equations on Hermitian manifolds.
The paper extends statistical estimation techniques under differential privacy.
problem Establishing sample complexity bounds for estimation tasks under differential privacy.
method Proposes analogues of Le Cam's method, Fano's inequality, and Assouad's lemma under central differential privacy.
result Optimal sample complexity bounds for discrete distribution estimation under total variation and ℓ2 distances. We give sharp C2,α estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge-Ampère equations with conical singularities. As an application…
We describe how to use the perturbation theory of Caffarelli to prove Evans-Krylov type C2,α estimates for solutions of nonlinear elliptic equations in complex geometry, assuming a bound on the Laplacian of the solution. Our results can be used to replace the various Evans-Krylov type arguments in the complex geom…
We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove C∞ a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced…
This paper proves Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
problem Establishing Hölder estimates on singular Kähler varieties.
method Geometric regularization based on partial C0 estimate. result Uniform Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
New complexity measure for interactive learning reduces regret to near-optimal levels.
problem Challenges in sample-efficient, adaptive learning algorithms for interactive decision making.
method Introduces the Decision-Estimation Coefficient and the Estimation-to-Decisions (E2D) principle.
result Unified algorithm design principle E2D achieves optimal sample-efficient learning.
Paper offers a framework for estimating symmetric properties efficiently.
problem Estimating symmetric properties of distributions from samples.
method General framework using profile maximum likelihood (PML) distribution.
result Optimal sample complexity for many properties, practical algorithms.
Study shows sample complexity for logistic regression with normal covariates.
problem Estimating parameters of logistic regression with normal design.
method Analyzes sample complexity in terms of dimension and inverse temperature.
result Shows two change-points in sample complexity curve based on inverse temperature.
VSE estimates complex processes from noisy measurements without a model.
problem Estimating states of complex, model-free processes from noisy data.
method Variational state estimation using recurrent neural networks (RNNs) in both learning and inference phases.
result VSE provides a competitive state estimate for a benchmark process (Lorenz system) compared to known and data-driven methods.
New estimate of semimeander complexity for knots with more than 10 crossings.
problem Estimating the complexity of semimeander diagrams of knots.
method Proved a new upper bound on the number of crossings for semimeander diagrams of knots with more than 10 crossings.
result For knots with more than 10 crossings, semimeander diagrams have no more than 0.31⋅1.558cr(K) crossings. We derive a priori C2 estimates for the χ-plurisubharmonic solutions of general complex Hessian equations with right-hand side depending on gradients.
Improved sample complexity for diffusion models without needing empirical risk minimizers.
problem Theoretical limitations in sample complexity for diffusion models.
method Structured decomposition of score estimation error, eliminating dependence on neural network parameters.
result Achieved sample complexity bound of O(ε^(-4)) without empirical risk minimizer access.
Deep learning improves causal effect estimation from complex observational data.
problem Estimating causal effects from complex observational data with low bias.
method Unified deep learning framework using multitask recurrent neural networks.
result Deep learning estimator shows lower bias in causal effect estimates.
Uniform estimates for complex Monge-Ampère equations on hermitian varieties.
problem Uniform boundedness of Chern-Ricci flat potentials in conifold transitions.
method Proving uniform a priori estimates for degenerate complex Monge-Ampère equations.
result Generalization of a theorem to hermitian contexts.
New findings show Rademacher complexities are not crucial for learning complexities.
problem Understanding the sample complexity of learning with squared loss in convex classes.
method Novel learning procedure combining mean estimation and Talagrand's generic chaining method.
result Sample complexity is determined by the limiting Gaussian process, not Rademacher complexities.
Adapts PDE method to prove L∞ estimates for complex Hessian equations.
problem Proving L∞ estimates for complex Hessian equations on transverse Kähler manifolds. method Adapts PDE approach of Guo-Phong-Tong and Guo-Phong-Tong-Wang [17, 18].
result Obtains L∞ estimate for transverse complex Monge-Ampère equations. New robust estimators achieve subgaussian bounds using VC-dimension.
problem Robust estimation of sparse and corrupted data.
method Use of VC-dimension to measure statistical complexity.
result First robust estimators for sparse estimation with subgaussian rate.
We study the cohomology with high tensor powers of Nakano q-semipositive line bundles on complex manifolds. We obtain the asymptotic estimates for the dimension of cohomology with high tensor powers of semipositive line bundles over q-convex manifolds and various possibly non-compact complex manifolds, in which the o…
New method estimates graphons from multiple networks with high accuracy and low complexity.
problem Estimating graphon function from multiple networks with different node sets and sizes.
method Histogram-based estimator that aligns nodes across all networks.
result High accuracy and low computational complexity achieved.
We study three fundamental statistical-learning problems: distribution estimation, property estimation, and property testing. We establish the profile maximum likelihood (PML) estimator as the first unified sample-optimal approach to a wide range of learning tasks. In particular, for every alphabet size k and desired…
Derives L∞ estimate for Kähler-Ricci flows with weaker conditions.
problem Estimating solutions to Kähler-Ricci flows under weaker conditions.
method Extends recent techniques to more general geometric cases.
result Derives L∞ estimate for Kähler-Ricci flows with weaker conditions. Derives estimates for geometric elliptic equations on complex manifolds.
problem Estimating solutions of geometric elliptic equations on complex manifolds.
method Derives a priori real Hessian estimates independent of the right-hand side.
result Establishes optimal C1,1 regularity of geometric envelopes. NanoFlow reduces parameter complexity in normalizing flows.
problem Efficient parameter complexity in flow-based models.
method Single neural density estimator with flow indication embedding.
result Sublinear parameter complexity achieved.
The paper addresses instability in KL divergence estimation using a neural network discriminator.
problem Unstable estimation of KL divergence due to discriminator complexity.
method Using a Reproducing Kernel Hilbert Space (RKHS) to control discriminator complexity.
result Theoretical bound on error probability of KL estimates based on discriminator complexity in RKHS.
Develops black-box methods to estimate parameters of complex models.
problem Lack of efficient methods to produce simulations for complex statistical models.
method Pre-training deep neural networks on extensive simulated databases for well-structured likelihoods. Iterative algorithm for other complex dependencies.
result Successfully estimates and quantifies uncertainty of parameters from non-Gaussian models.
In sparse Bayesian learning (SBL), Gaussian scale mixtures (GSMs) have been used to model sparsity-inducing priors that realize a class of concave penalty functions for the regression task in real-valued signal models. Motivated by the relative scarcity of formal tools for SBL in complex-valued models, this paper propo…
Estimates for complex Hessian equations on Hermitian manifolds.
problem Establishing estimates for solutions to complex Hessian equations.
method Using concavity inequality for complex sum-of-Hessian operators.
result Second-order estimates for admissible solutions on Hermitian manifolds.
In this note, a gradient estimate for the complex Monge-Ampere equation is established. It differs from previous estimates of Yau, Hanani, Blocki, P. Guan, B. Guan - Q. Li in that it is pointwise, and depends only on the infimum of the solution instead of its C0 norm.