Investigates regularity of solutions to complex Hessian equation.
problem Regularity of solutions to complex Hessian equation.
method Analyzes solutions to Dirichlet problem with specific density condition.
result Establishes conditions for regularity of solutions.
New metric for probability measures connects physics and geometry.
problem Developing a new metric for probability measures.
method Transport Hessian metric, formulated dynamical systems.
result Connections to physics equations and mathematical models.
Study currents from semi-convex functions, apply to Hessian measures.
problem Understanding currents from semi-convex functions.
method Analyze integer multiplicity rectifiable currents from subgradient graphs of semi-convex functions.
result Weak continuity theorem for currents with pointwise convergence.
Study on existence and properties of continuous solutions to complex Hessian equations.
problem Existence and properties of continuous solutions to complex Hessian equations.
method Established new capacity estimates and weak stability estimates for the m m m -Hessian measure. result Existence of continuous solutions to the complex Hessian equation under certain conditions.
New algorithms estimate Hessians using random directions for faster stochastic optimization.
problem Efficiently estimating Hessians for stochastic optimization.
method Generalized Hessian estimators using random directions and noisy function measurements.
result Asymptotically unbiased estimators with lower bias for more measurements.
Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.
problem Solving Monge-Ampère equations on compact Hessian manifolds.
method Perron method, compactness properties of normalized quasi-convex functions, local and global comparison principles for twisted Monge-Ampère operators.
result Solves Monge-Ampère equations involving arbitrary probability measures.
Theory developed for complex Hessian measures on Hermitian manifolds.
problem Defining and analyzing complex Hessian measures on Hermitian manifolds.
method Potential theory for m-subharmonic functions with respect to a Hermitian metric.
result Equivalence between polar sets and negligible sets for m-subharmonic functions.
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…
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.
The paper introduces a Hessian-based method to improve generalization in fine-tuned deep neural networks.
problem Improving generalization in fine-tuned deep neural networks, especially in noisy conditions.
method PAC-Bayesian analysis to identify a Hessian-based distance measure, proving generalization bounds, and developing an algorithm with a generalization error guarantee.
result Hessian-based distance measure correlates well with observed generalization gaps and can match the scale of these gaps in practice.
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.
New findings challenge the use of flatness measures in neural networks.
problem The validity of flatness measures in assessing generalization in neural networks.
method Analysis of Hessian-based flatness norms and their relation to generalization.
result Solutions with large weights and low loss are often sharper than expected, contradicting flatness measures.
Unified approach to domain generalization by aligning gradients and Hessians.
problem Developing models that generalize well across unseen domains.
method Moment Alignment, extending transfer measure to DG, aligning derivatives across domains.
result Moment Alignment unifies gradient and Hessian matching approaches, improving generalizability.
Study solves complex Hessian equations with prescribed singularities on compact Kähler manifolds.
problem Solving complex Hessian equations with specific singularity types on compact Kähler manifolds.
method Analyzes the total mass of complex Hessian measures and solves equations with prescribed singularities.
result Proves non-decreasing total mass of complex Hessian measures and solves complex Hessian equations.
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.
Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.
problem Interior Hessian estimates for solutions with prescribed Lipschitz phases.
method Allard-type regularity theorem, geometric measure theory, geometry of Lagrangian graphs, De Giorgi-Nash-Moser iteration.
result Sharp interior Hessian estimates for solutions with critical and supercritical phases.
We introduce a scalable measure of curvature for analyzing training dynamics of large language models.
problem Analyzing the training dynamics of large language models due to high computational cost of measuring Hessian sharpness.
method We introduce critical sharpness and relative critical sharpness as computationally efficient measures capturing Hessian sharpness phenomena.
result We provide the first demonstration of sharpness phenomena at scale up to 7B parameters.
The paper describes a fine representation of the Ricci tensor and Hessian on RCD spaces.
problem Understanding the structure of the Ricci tensor and Hessians on RCD spaces.
method Polar decomposition of the Ricci tensor and Hessians, providing regularity results.
result The Ricci tensor and Hessians on RCD spaces can be represented by a polar decomposition, revealing their regularity.
CWGD measures gradient diversity weighted by curvature, improving SGD convergence.
problem Gradient noise in high-curvature directions is underestimated by standard methods.
method CWGD weights gradient diversity by the inverse square root of the Hessian.
result CWGD-Cosine reduces optimization error by up to 20% compared to standard cosine annealing.
This work connects the Hessian to the decision boundary complexity in neural networks.
problem Understanding the decision boundary complexity in high-dimensional input space.
method Characterizing the decision boundary using the Hessian top eigenvectors and analyzing the number of outliers.
result The number of outliers in the Hessian spectrum is proportional to the complexity of the decision boundary.
Develops calculus for tamed Dirichlet spaces using measure theory.
problem Defines calculus for measure spaces with Dirichlet forms.
method Introduces first and second order calculus on tamed Dirichlet spaces.
result Defines various geometric objects on tamed Dirichlet spaces.
Study of complex Hessian equations using subharmonic functions and geodesics.
problem Understanding geodesics within complex Hessian equations.
method Perron envelope construction, comparison principle, rooftop equality, Kiselman minimum principle.
result Established criterion for geodesic connectivity among m m m -subharmonic functions. PWGF escapes saddle points in nonconvex optimization.
problem Escaping saddle points in nonconvex optimization.
method PWGF uses noisy perturbations via Gaussian process to escape saddle points.
result PWGF achieves second-order optimality for nonconvex objectives.
New structure in neural network Hessians explains outliers, improving subspace approximation.
problem Explaining outliers in the spectrum of deepnet Hessians.
method Identified a two-level structure in the Hessian, showing it's not a covariance but a second moment matrix.
result Shows the means of gradients have an additive two-way structure, leading to outliers in the spectrum.
Paper proposes a new flatness measure for neural networks to improve generalization.
problem Generalization in deep learning models, especially with overparameterization.
method Soft rank measure of the Hessian to assess flatness and generalization.
result Soft rank flatness measure accurately estimates generalization gaps for various models.
Hessian alignment improves OOD generalization in deep learning.
problem Improving deep learning models' ability to generalize to out-of-distribution data.
method Analyzed Hessian and gradient alignment for domain generalization using recent OOD theory.
result Hessian alignment methods achieve promising performance on various OOD benchmarks.
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.
New flatness measure for deep networks invariant to scaling.
problem Lack of invariant flatness measures for deep networks under parameter rescaling.
method Introduced a quotient manifold structure and Hessian-based invariant flatness measure.
result Confirms that Large-Batch SGD minima are sharper than Small-Batch SGD minima.
Our work connects parameter magnitudes and Hessian eigenspaces in deep neural nets.
problem Understanding the relationship between parameter magnitudes and Hessian curvature in deep learning models.
method Developed a matrix-free algorithm based on sketched SVDs to measure similarity between parameter masks and Hessian eigenspaces.
result Top Hessian eigenvectors tend to be concentrated around larger parameters, indicating a connection between parameter magnitudes and loss curvature.
The paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.
problem Discrete construction of Hessian and divdiv complexes on triangulations.
method Construction of discrete Hessian and divdiv complexes using finite elements and Dirac measures on triangulations.
result The cohomology of the constructed complexes is isomorphic to the continuous de Rham cohomology.
Researchers introduce new functionals to measure distance from Kähler-Einstein metrics.
problem Estimating how close a metric is to Kähler-Einstein.
method Introducing Ricci-Calabi and H-functionals, and proving moment weight inequalities and Hessian formulas.
result Established inequalities and formulas to measure distance and conditions for existence of Kähler-Einstein metrics.
Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.
problem Finding capillary convex bodies with prescribed k k k -th capillary area measure. method Solving a Hessian-type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution under natural conditions.
Noise injection regularizes Hessian, improving neural network training and generalization.
problem Regularizing over-parameterized neural networks with nonconvex and nonlinear geometry.
method Injecting isotropic Gaussian noise into weight matrices and designing a two-point estimate of the Hessian penalty.
result Effective regularization of Hessian improves generalization, achieving up to 2.4% test accuracy increase.
Introduces HTV to measure function complexity in learning schemes.
problem Assessing the complexity of supervised-learning schemes.
method Defines Hessian-Schatten total variation (HTV) as a seminorm to quantify function complexity.
result HTV is invariant to rotations, scalings, and translations, and its minimum value is achieved for linear mappings.
Enhances influence functions for deep models without costly Hessian inversion.
problem Challenges in computing Hessian matrix inverse for deep models.
method Substitutes Hessian inverse with identity matrix and extends applications.
result TracIn method performs well in measuring model utility and fairness.
We prove Obata's rigidity theorem for metric measure spaces that satisfy a Riemannian curvature-dimension condition. Additionally, we show that a lower bound K K K for the generalized Hessian of a sufficiently regular function u u u holds if and only if u u u is K K K -convex. A corollary is also a rigidity result for higher or…
The paper proves a Gaussian measure version of the Brunn-Minkowski inequality.
problem Proving a Brunn-Minkowski inequality for Gaussian measure.
method Raywise radial-tangential localization of the Hessian energy of a solution to a Neumann problem.
result The largest number α_γ(n) for the inequality is found to be 1 - 2/(n-1) * (Γ(n/2)^2 / Γ((n-1)/2)^2).
Improved graph neural network bounds using graph diffusion matrix.
problem Empirical performance of graph neural networks on real-world graphs.
method Unified model of graph neural networks, focusing on feature diffusion matrix stability.
result Generalization bounds scale with largest singular value of feature diffusion matrix, smaller than prior bounds.
In this paper, we extend the DC Calculus introduced by Perelman on finite dimensional Alexandrov spaces with curvature bounded below. Among other things, our results allow us to define the Hessian and the Laplacian of DC functions (including distance functions as a particular instance) as a measure-valued tensor and a …
New method proves asymptotic normality for matrix sensing problems.
problem Proving asymptotic normality for matrix sensing under general convex losses.
method Riemannian geometry to handle degeneracy of the Hessian due to rotational symmetry.
result Proves n ( φ 0 − φ ∗ ) → D N ( 0 , ( H ∗ ) − 1 ) \sqrt{n}(φ^0-φ^*)\xrightarrow{D}N(0,(H^*)^{-1}) n ( φ 0 − φ ∗ ) D N ( 0 , ( H ∗ ) − 1 ) as n o ∞ n o\infty n o ∞ . URGE improves diffusion model quality without gradients or Hessian.
problem Improving sample quality in diffusion models without gradient evaluations.
method Path-wise importance reweighting via Girsanov change of measure.
result URGE achieves better generation quality than existing methods.
The study proves that certain noncompact Hessian manifolds are diffeomorphic to R^n.
problem Characterizing complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature.
method Using a geometric flow on noncompact affine Riemannian manifolds, constructing Hessian metrics, and proving diffeomorphism.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature are diffeomorphic to R^n if their tangent bundle has maximal volume growth.
Paper solves Minkowski problem for k-torsional rigidity.
problem Solving Minkowski problem for k-torsional rigidity.
method Constructing Hadamard variational formula, proposing k-torsional measure, using curvature flow method.
result Existence of smooth non-even solutions to the Minkowski problem.
Abstract: Survey on quadratic Hessian equations, their properties, and open problems.
problem Understanding quadratic Hessian equations and their solutions.
method Survey and review of existing research.
result Survey of entire solutions, viscosity solutions, and Hessian estimates.
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
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.
Characterizes and describes selfsimilar Hessian manifolds with homothetic vector fields.
problem Understanding the structure and properties of selfsimilar Hessian manifolds.
method Characterization and description of selfsimilar manifolds with homothetic vector fields.
result Any selfsimilar Hessian manifold with a potential homothetic vector field is locally isomorphic to a product of radiant Hessian manifolds.
The paper studies the loss landscape of regularized deep matrix factorization, revealing unique and sharp minimizers.
problem Understanding the loss landscape and minimizers of regularized deep matrix factorization problems.
method Theoretical analysis of ℓ 2 \ell^2 ℓ 2 -regularized deep matrix factorization/deep linear network training problems with squared-error loss. result The unique end-to-end minimizer exists for all target matrices except for a set of Lebesgue measure zero.