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.
Study estimates eigenvalues for concave Hessian operators on convex domains.
problem Estimating eigenvalues for concave elliptic Hessian operators.
method Investigates Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators.
result Existence and properties of the first nonzero eigenvalue and eigenfunction.
Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.
problem Eigenvalue problem for complex Hessian operator on pseudoconvex manifolds.
method Established C1,1-regularity and uniqueness of the first eigenfunction, derived variational formula for the first eigenvalue. result Derivation of a bifurcation-type theorem and geometric bounds for the eigenvalue.
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.
New method shows Hessian estimator from random samples converges to true Hessian on complex manifolds.
problem Uncertainty in Hessian estimator accuracy on complex manifolds with boundaries and nonuniform sampling.
method Locally fitting quadratic polynomials, rigorous theoretical analysis under mild conditions.
result The Hessian estimator asymptotically converges to the true Hessian, even near boundaries.
New algorithm adds Hessian regularization to improve neural network robustness.
problem Improving neural network robustness against adversarial attacks.
method Proposes an efficient algorithm to train neural networks with Hessian operator-norm regularization.
result Hessian operator-norm regularization increases neural network robustness over input gradient regularization.
Introduces Floer functions and Floerfolds for intrinsic properties.
problem Complex transformation of Hessian under chart transition.
method Introduces Floer functions and Floerfolds to address intrinsic properties.
result Floer functions and Floerfolds provide intrinsic conditions for Hessian.
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.
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.
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.
A new spline method for manifold learning using Hessian-based curvature penalties.
problem Learning manifolds with curvature penalties in high dimensions.
method Generalizes thin-plate splines to flat manifolds using Hessian matrices, minimizing square error with curvature constraints.
result Existence and uniqueness of the spline solution, expressed as Green's functions and Hessian approximations.
Proof that certain complex equations have only simple solutions.
problem Characterizing solutions to complex equations.
method Analyzing fully nonlinear elliptic operators.
result Entire smooth solutions are quadratic polynomials.
On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group O(n) by using only Euclidean coordinates …
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.
Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.
problem Computing Laplace-Beltrami on constrained submanifolds.
method Embedded gradient vector field method, explicit formula derivation.
result Explicit formula for Laplace-Beltrami on orthogonal group.
We analyze the Hessian spectra of large models up to 100B parameters.
problem Accurate Hessian spectra of large foundation models are difficult to obtain.
method We use shard-local finite-difference Hessian vector products and stochastic Lanczos quadrature.
result We produce the first large-scale spectral density estimates of foundation models.
Paper derives estimates for Hessian equations under concavity assumptions.
problem Estimating solutions to Hessian equations with concavity constraints.
method Exploiting concavity of sums of Hessian operators, deriving Pogorelov estimates.
result Entire admissible solutions in R^n with quadratic growth are quadratic polynomials.
Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.
problem Problems posed by Delanoë and Urbas related to Hessian quotient equations.
method Maximum principle argument and new test function introduced to prove estimate.
result Unobstructed second order a priori estimate for real Hessian quotient equation in 2D.
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.
The paper introduces a new differential-geometric system which originates from the theory of m-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise to the following absolute geometric invariants: invariant derivatives of the surf…
Study Hessian equations on compact Kähler manifolds with prescribed singularities.
problem Characterize finite energy ranges of the Hessian operator and solutions of degenerate complex Hessian equations.
method Reformulate pluripotential results to Hessian setting and use a new method.
result Prove solutions of degenerate complex Hessian equations have the same singularity type as the model potential.
Study on manifolds with density using modified Hessians for curvature comparison.
problem Developing comparison geometry on manifolds with density.
method Modified Hessian approach based on weighted sectional curvature framework.
result Derivation of Hessian comparison and shape operator comparison theorems.
New insights into Hessian structure of neural networks reveal two forces.
problem Understanding the Hessian structure of neural networks.
method Analyzing the static and dynamic forces, comparing limit distributions using random matrix theory.
result The Hessian structure arises from a combination of static and dynamic forces, with C being a primary driver. 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.
New Bethe-Hessian method improves community detection in sparse networks.
problem Detect communities in sparse networks efficiently.
method Spectral clustering using the Bethe-Hessian matrix.
result Bethe-Hessian consistently estimates block number above Kesten-Stigum threshold.
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.
Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
We consider deep classifying neural networks. We expose a structure in the derivative of the logits with respect to the parameters of the model, which is used to explain the existence of outliers in the spectrum of the Hessian. Previous works decomposed the Hessian into two components, attributing the outliers to one o…
New formulas for Riemannian gradient and Hessian on manifold metrics.
problem Evaluate Riemannian gradient and Hessian for various metrics on manifolds.
method Explicit formulas derived from Levi-Civita connection and projection.
result Derives new metrics and optimization frameworks on manifolds.
Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the n…
Large batch size training of Neural Networks has been shown to incur accuracy loss when trained with the current methods. The exact underlying reasons for this are still not completely understood. Here, we study large batch size training through the lens of the Hessian operator and robust optimization. In particular, w…
We investigate the short-time expansion of the heat kernel of a Laplace type operator on a compact Riemannian manifold and show that the lowest order term of this expansion is given by the Fredholm determinant of the Hessian of the energy functional on a space of finite energy paths. This is the asymptotic behavior to …
New stability conditions for ZO methods reveal unique regularization effects.
problem Understanding optimization dynamics of ZO methods in deep learning.
method Explicit step size conditions and stability bounds derived for ZO methods.
result ZO methods operate near the edge of stability, with regularization effects specific to Hessian trace vs. eigenvalue.
The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.
problem Generalizing differential geometric structures to algebroids.
method Introducing statistical, conjugate connection, and Hessian structures on anti-commutable pre-Leibniz algebroids.
result Statistical and conjugate connection structures are equivalent for admissible connections.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. We model how Lipschitz continuity changes during neural network training.
problem Understanding how Lipschitz continuity evolves during training.
method We use a system of stochastic differential equations to capture the dynamics of Lipschitz continuity under SGD.
result We identify three factors driving the evolution of Lipschitz continuity: gradient flow projection, gradient noise, and Hessian projection.
The volume distance from a point p to a convex hypersurface M of the (N+1)-dimensional space is defined as the minimum (N+1)-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane H(p) we ob…
We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a lower bound independent of the dimension. We also estimate the gap wh…
Let L be an ample bundle over a compact complex manifold X. Fix a Hermitian metric in L whose curvature defines a Kähler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D on functions which arises in the study of scalar curvature. We quantise D by the Hessian E(k) of balancing energy, a f…
Large learning rates cause parameter instability, leading to better generalization.
problem Understanding why deep neural networks perform well despite operating outside the traditional stability regime.
method Analyzing the effect of large learning rates on the orientation of Hessian eigenvectors and parameter exploration.
result Large learning rates induce parameter instability, leading to better generalization through exploration of flatter regions of the loss landscape.
The existence and continuity for the Calderon projector of the perturbed odd signature operator on a 3-manifold is established. As an application we give a new proof of a result of Taubes relating the mod 2 spectral flow of a family of operators on a homology 3-sphere with the difference in local intersection numbers o…
SVRN accelerates Newton methods by reducing variance and improving performance.
problem Improving the efficiency of Newton methods for large-scale optimization problems.
method Stochastic Variance-Reduced Newton (SVRN) algorithm that accelerates Subsampled Newton and Iterative Hessian Sketch algorithms.
result SVRN accelerates Newton methods by reducing the number of passes over the data, achieving a significant improvement in performance.
New method speeds up Bayesian inverse problem solving with neural operators.
problem Solving infinite-dimensional Bayesian inverse problems with high computational cost.
method Delayed-acceptance geometric MCMC driven by derivative-informed neural operator surrogates.
result Significant speedup in generating posterior samples (3-9 times faster).
New method constructs solution operators for PDEs with prescribed support properties.
problem Constructing solution operators for under/overdetermined PDEs with specific support properties.
method Using a recovery on curves condition and taking smooth averages over curves, we obtain integral solution operators and representation formulas.
result Our method leads to integral representation formulas for overdetermined PDEs and solution operators for underdetermined PDEs.
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.
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.
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.