Paper proves inequalities on Hermitian manifolds with applications to bounded solutions.
problem Establishing mixed Hessian inequalities on Hermitian manifolds.
method Weak convergence theorem of complex Hessian operators and general mixed Hessian inequality.
result Existence of bounded solutions of complex Hessian equations.
We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge-Ampère equation. Exploiting this and using Kołodziej's mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact…
Study solves complex Hessian equation on Hermitian manifolds.
problem Solving Hessian equations on Hermitian manifolds with mixed structure.
method Derive a priori estimates and solve Dirichlet problem under conditions.
result Solvability of the Dirichlet problem for mixed Hessian equations.
Proves a generalized Minkowski inequality for starshaped domains.
problem Proving a generalized Minkowski inequality for smooth, (k−1)-convex starshaped domains. method Solvability of the degenerate k-Hessian equation on the exterior domain Rn∖Ω. result Generalized Minkowski inequality holds for smooth, (k−1)-convex, starshaped domains. Paper discusses solving generalized Hessian inequalities with various operators.
problem Finding global solutions to generalized Hessian inequalities.
method Analyzes various Hessian operators and provides conditions for global solvability.
result Provides necessary and sufficient conditions for global solvability of generalized Hessian inequalities.
We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from polytopes in R^n specified by m inequalities. The walk is a discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold equipped with the metric induc…
Derives concavity inequality and estimates for k-Hessian equations.
problem Interior estimates and curvature estimates for k-Hessian equations. method Concavity inequality and semi-convexity condition.
result Interior estimates and Liouville-type result for semi-convex solutions.
HMC with leapfrog integrator mixes faster than MALA under certain smoothness conditions.
problem Analyzing the mixing time of HMC and MALA for sampling from smooth distributions.
method Bounding gradient complexity and leveraging invariance of joint distribution.
result Metropolized HMC with more leapfrog steps outperforms MALA in total variation distance.
New Hessian estimates for heat equations on manifolds.
problem Estimating Hessian matrices for heat-type equations on Riemannian manifolds.
method Using Bismut-Stroock Hessian formula, with explicit coefficients and delay/growth rate functions.
result Novel backward weak Harnack inequality and precise pointwise Hessian estimates for eigenfunctions.
Establishes a concavity property for positive Hessian quotient operators.
problem Analyzing positive Hessian quotient operators on Riemannian manifolds.
method Proves a special concavity property and a Jacobi inequality.
result Proves a Jacobi inequality for symmetric tensors.
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.
Characterizes complex Hessian equations for bounded energy functions.
problem Understanding degenerate complex Hessian equations for bounded energy functions.
method Proving sublevel set estimates and using Sobolev inequalities.
result Characterization of degenerate complex Hessian equations for bounded (p,m)-energy functions. Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.
problem Christoffel-Minkowski problem and Hessian equations under rotational symmetries.
method Constructing explicit convex solutions to mixed Monge-Ampère equations on \(\mathbb{R}^n\) under radial symmetry.
result Explicit representation formula for the support function of the resulting convex body.
Paper estimates curvature of semi-convex solutions in hyperbolic space.
problem Curvature estimation for semi-convex solutions in hyperbolic space.
method Used concavity inequality for Hessian operator.
result Established curvature estimates for semi-convex solutions and admissible solutions.
We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes…
The paper proves a Liouville theorem for special Lagrangian equations with convexity conditions.
problem Proving Liouville theorems for special Lagrangian equations with specific conditions.
method Using Neumann-Poincaré inequality, mean value inequality for superharmonic functions, and geometric measure theory.
result Derives global and interior Hessian estimates for solutions of special Lagrangian equations.
In this paper, we studied integrals involving both real and complex Hessian operators over bounded domain. Poincare type inequalities were proved in both cases which generalized a early results of Trudinger and Wang.
RHMC improves sampling polytopes defined by inequalities with barriers.
problem Sampling polytopes defined by inequalities efficiently.
method Riemannian Hamiltonian Monte Carlo (RHMC) with a hybrid of Lewis weights and logarithmic barriers.
result RHMC achieves mixing rate of ildeO(m1/3n4/3) for polytopes defined by m inequalities in Rn. In this paper, we introduce several mixed Lp geominimal surface areas for multiple convex bodies for all p=−n. Our definitions are motivated from an equivalent formula for the mixed p-affine surface area. Some properties, such as the affine invariance, for these mixed Lp geominimal surface areas are prove…
The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.
problem Estimating solutions to nonlinear weighted parabolic equations.
method Derives Li-Yau and Hamilton type gradient estimates, and Hessian estimates.
result New gradient and Hessian estimates for positive solutions of nonlinear parabolic equations.
Paper solves overdetermined k-Hessian equation in exterior domains.
problem Overdetermined problem for k-Hessian equation in exterior domains. method Combining integral identities and geometric inequalities, derived general monotone formulas.
result Established general monotone formulas for k-admissible solutions. Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.
problem Geometric inequalities and mixed Hodge-Riemann relations for translation-invariant valuations.
method Proves mixed Hodge-Riemann relations for various convex bodies and their mixed volumes.
result Strengthened geometric inequalities for lower dimensional convex bodies.
The paper addresses geometric analysis on non-compact Riemannian manifolds, proving Calderón-Zygmund inequalities.
problem Proving Calderón-Zygmund inequalities on non-compact Riemannian manifolds without positive injectivity radius.
method Probabilistic tools, Hessian formulas, and Bismut type representations for heat semigroups.
result The paper proves the Calderón-Zygmund inequality for 1<p<2 under a lower Ricci curvature bound, and for p>2 under additional curvature conditions. Recently, the first named author together with Xinan Ma \cite{ma2015neumann}, have proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence of classical Neumann problems under the uniforml…
New inequalities generalize Li's theorem on mixed Hodge structures.
problem Generalizing Li's theorem on mixed Hodge structures.
method Develop new Hodge-Riemann bilinear relations in mixed settings.
result New Khovanskii-Teissier type inequalities and log-concavity results.
Paper develops estimates for Lagrangian phase changes in 2D.
problem Interior estimates for Lagrangian phase changes in 2D.
method Modified doubling technique to handle degenerate Jacobi inequalities.
result Interior Hessian and gradient estimates established for critical phase.
Improved privacy-preserving linear regression via iterative Hessian mixing.
problem Differentially private linear regression with improved accuracy and efficiency.
method Iterative Hessian Mixing (IHM) for differentially private ordinary least squares (DP-OLS).
result IHM provides better utility guarantees and outperforms AdaSSP in empirical evaluations.
The paper develops new inequalities for Markov chain sums, linking them to mixing time.
problem Establishing concentration inequalities for Markov chain sums.
method Developed novel concentration inequalities for geometrically ergodic Markov chains, linking bounds to mixing time constants.
result Explicit bounds for additive functionals of Markov chains, linked to Rosenthal inequality constants and mixing properties.
The paper explores dualities in differential equations and their applications in Riemannian geometry.
problem Developing comparison theorems for mixed type differential equations.
method Utilizing dualities in differential equations and inequalities, and applying them to Riemannian geometry.
result Proves Hessian and Laplacian comparison theorems under various curvature assumptions.
Paper doubles Hessian estimates for special Lagrangian equation with constraints.
problem Estimating Hessian for special Lagrangian equation under general phase constraints.
method Doubling argument, Alexandrov-type theorems.
result Established Hessian estimates for special Lagrangian equation.
Proves Hessian estimates for special Lagrangian equation with new proofs.
problem Interior Hessian estimates for special Lagrangian equation
method Doubling proofs, higher codimension analogue of previous methods
result Higher codimension analogue of gradient estimate for minimal hypersurfaces
Let (X,ω) be a compact Kähler manifold of dimension n and fix 1≤m≤n. We prove that the total mass of the complex Hessian measure of ω-m-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge i…
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.
Survey of global Calderón-Zygmund inequalities on Riemannian manifolds.
problem Validity and failure of W2,p regularity for Poisson equation solutions. method Various geometric conditions and methods to obtain Lp-Hessian estimates. result Integral inequality may fail even with lower sectional curvature bound.
Improved score matching methods for estimating score functions and Hessians without high dimensionality.
problem Estimating score functions and Hessians efficiently in high-dimensional data.
method Implicit score matching and denoising score matching, leveraging Gagliardo-Nirenberg inequalities.
result Achieves convergence rates similar to denoising score matching and estimates Hessians without dimensionality issues.
At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes…
We generalize an inequality of E. Heintze and H. Karcher [8] for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for k-Ricci curvature. Even in the case of a pointwise bound, this generalizes the classical inequality by replacing a sectional curvature bound with a k-Ricci b…
The paper analyzes convergence rates of Langevin dynamics and Proximal Sampler using Φ-divergence.
problem Analyzing convergence rates of Langevin dynamics and Proximal Sampler.
method Extending mixing time analyses to Φ-divergence, using strong data processing inequalities. result Convergence of Φ-divergence to 0 exponentially fast along Unadjusted Langevin Algorithm and Proximal Sampler. New criterion for solving inverse Hessian equations, including J-equation.
problem Existence of solutions to inverse Hessian equations, including J-equation.
method Stability of pairs in the sense of Paul, formulated in terms of GIT criterion.
result New numerical criterion for existence of solutions to inverse Hessian equations.
The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.
problem Finding extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
method Characterizations obtained under suitable geometric constraints.
result Characterizations of extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz φ-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz φ-radial addition of two star bodies, we derive a f…
Mirror Langevin Algorithm converges with zero bias.
problem Achieving convergence with zero bias in discrete-time sampling.
method Discretization of Mirror Langevin Diffusion and mean-square analysis.
result Mirror Langevin Algorithm converges with zero bias.
Rapid mixing of Langevin dynamics on Riemannian manifolds
problem Mixing time of Langevin dynamics on Riemannian manifolds
method Relation between Langevin processes in domain and image
result Achievable polynomial mixing times
Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
MALA mixes efficiently under smoothness and isoperimetry assumptions.
problem Sampling from target densities efficiently.
method Metropolis-Adjusted Langevin algorithm (MALA) with smoothness and isoperimetry assumptions.
result MALA mixes in $O\left(\frac{(LΥ)^{\frac12}}{ψ_μ^2} \log\left(\frac{1}ε
ight)
ight)$ iterations.
We establish variational formulas for Ricci upper and lower bounds, as well as a derivative formula for the Ricci curvature. As applications, constant curvature manifolds, Einstein manifolds and Ricci parallel manifolds are identified, respectively, with different integral-differential formulas and semigroup inequaliti…
This paper investigates the supervised learning problem with observations drawn from certain general stationary stochastic processes. Here by \emph{general}, we mean that many stationary stochastic processes can be included. We show that when the stochastic processes satisfy a generalized Bernstein-type inequality, a u…
The paper proves Liouville rigidity for Hessian equations, characterizing geometric conditions for constant solutions.
problem Characterizing geometric conditions for constant solutions in Hessian equations.
method Recursive geometric condition (Liouville admissibility) and anisotropic constructions.
result The Liouville-type property is characterized as a geometric property of the admissible set.