The paper studies efficient Hessian fitting methods for stochastic optimization.
problem Efficient Hessian fitting for stochastic optimization.
method Preconditioned Stochastic Gradient Descent (PSGD) method and Lie groups.
result Hessian fitting problem is strongly convex in certain Lie groups.
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.
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.
Analyzes the Hessian of ReLU networks, proving skewed eigenvalue distribution.
problem Characterizing the Hessian at spurious minima in shallow ReLU models.
method Symmetry breaking and representation theory techniques.
result Proves skewed eigenvalue distribution of Hessian at spurious minima.
Survey of SDR methods for high-dimensional regression and embedding.
problem Reducing dimensionality in high-dimensional data.
method Involves both statistical and machine learning approaches, covering inverse and forward regression methods.
result Supervised Kernel Dimension Reduction is equivalent to supervised PCA.
Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce …
A new screening rule improves lasso solving speed.
problem Efficiently solving lasso problems with high correlation.
method Uses second-order information from the Hessian to screen predictors.
result Outperforms alternatives on simulated and real data.
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.
Proposes a method for inference in high-dimensional classification with non-differentiable surrogate losses.
problem Lack of inference procedures for identifying driving factors in high-dimensional classification with non-differentiable surrogate losses.
method Kernel-smoothed decorrelated score and cross-fitted version for hypothesis tests and interval estimators.
result Valid and superior inference methods for high-dimensional classification with non-differentiable surrogate losses.
New preconditioners speed up SGD on Lie groups.
problem Accelerating SGD convergence on Lie groups.
method Online updating of low-rank and matrix-free preconditioners.
result Preconditioners maintain symmetry and invariance.
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.
This paper presents a new optimisation approach to train Deep Neural Networks (DNNs) with discriminative sequence criteria. At each iteration, the method combines information from the Natural Gradient (NG) direction with local curvature information of the error surface that enables better paths on the parameter manifol…
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.
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.
Deep (neural) networks have been applied productively in a wide range of supervised and unsupervised learning tasks. Unlike classical machine learning algorithms, deep networks typically operate in the \emph{overparameterized} regime, where the number of parameters is larger than the number of training data points. Con…
In this paper, we empirically study models for pricing Italian sovereign bonds under a reduced form framework, by assuming different dynamics for the short-rate process. We analyze classical Cox-Ingersoll-Ross and Vasicek multi-factor models, with a focus on optimization algorithms applied in the calibration exercise. …
Study calculates Hessian of Busemann function on Damek-Ricci spaces.
problem Calculating Hessian of Busemann function on Damek-Ricci spaces.
method Calculates eigenvalues of Hessian and proves positive definiteness.
result Hessian of Busemann function is positive definite.
The paper shows infinitely many components in Floer Hessians space.
problem Understanding the structure of Floer Hessians.
method Proving the existence of infinitely many connected components.
result Proves infinitely many connected components in Floer Hessians space.
The paper describes flat Hessian metrics on surfaces and their potentials.
problem Understanding Hessian metrics on surfaces.
method Theoretical description and explicit construction using integrable systems.
result Explicit construction of potentials for flat Hessian metrics on surfaces.
The paper quantizes Hessian structures on R^2 using KV-algebras.
problem Quantizing Hessian structures on a 2D space.
method Deformation quantization within Koszul-Vinberg algebras.
result Established links between deformation theory and Hessian geometry.
We prove that, in dimensions greater than 2, the generic metric is not a Hessian metric and find a curvature condition on Hessian metrics in dimensions greater than 3. In particular we prove that the forms used to define the Pontryagin classes in terms of the curvature vanish on a Hessian manifold. By contrast all anal…
Superintegrable systems on curved manifolds found to have Hessian structures.
problem Characterizing superintegrable systems on curved manifolds.
method Identifying and computing Hessian coordinates for superintegrable systems.
result Examples of superintegrable systems in 2D and 3D have natural Hessian coordinates.
Curved Frobenius manifolds link to Hessian metrics in geometry.
problem Understanding curved Frobenius manifolds and their relation to Hessian metrics.
method Analyzing the relationship between curved Frobenius structures and Hessian metrics on spaces with non-vanishing curvature.
result Consistent curved Frobenius structures on constant curvature spaces are linked to Hessian metrics.
New rigidity results for a generalized Ricci-Hessian equation on manifolds.
problem Understanding rigidity in generalized Ricci-Hessian equations on manifolds.
method Proving new rigidity results related to a generalized Ricci-Hessian equation.
result New rigidity results for the generalized Ricci-Hessian equation on Riemannian manifolds.
Study classifies 3D Hessian manifolds, proving their topology.
problem Global topology of 3D Hessian manifolds.
method Proved structure and analyzed Betti numbers.
result Complete topological classification of 3D Hessian manifolds.
New method improves Gaussian Mixture Model fitting speed.
problem Slow convergence of EM algorithm for complex data.
method Riemannian Newton Trust-Region method for Gaussian Mixture Models.
result Outperforms existing methods in runtime and iterations.
Criterion for solvability of complex 2-Hessian equation on compact Kähler manifolds.
problem Solvability of complex 2-Hessian equation on compact Kähler manifolds.
method Nakai--Moishezon-type criterion associated with the complex 2-Hessian equation.
result Criterion equivalent to existence of a smooth 2-admissible representative in complex dimension three.
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.
EGMU optimizes portfolios using KL divergence, ensuring positive solutions.
problem Constructing multi-factor target-exposure portfolios efficiently and accurately.
method Convex optimization framework minimizing KL divergence, with explicit solvers.
result Established feasibility and uniqueness of strictly positive solutions under convex-hull conditions.
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.
Derives Hessian estimates for Lagrangian mean curvature equation.
problem Lagrangian mean curvature equation with supercritical phase and bounded second derivatives.
method Derives a priori interior Hessian estimates.
result Hessian estimates for Lagrangian mean curvature equation.
A new unbiased Hessian estimator for expectation-based objectives.
problem Estimating Hessian for objectives with non-reparameterizable nodes.
method GO Hessian estimator for expectation-based objectives.
result GO Hessian provides unbiased and low-variance estimation of Hessian.
Constructs homogeneous Kähler structures on tangent bundles of Hessian manifolds.
problem Creating Kähler structures on tangent bundles of Hessian manifolds.
method Endowing Hessian manifolds with Kähler structures using group actions and homothetic vector fields.
result Homogeneous conformally Kähler structures on tangent bundles of selfsimilar Hessian manifolds.
This paper presents an improvement to model learning when using multi-class LogitBoost for classification. Motivated by the statistical view, LogitBoost can be seen as additive tree regression. Two important factors in this setting are: 1) coupled classifier output due to a sum-to-zero constraint, and 2) the dense Hess…
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.
A selfsimiar manifold is a Riemannian manifold (M,g) endowed with a homothetic vector field ξ. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…
This paper uncovers the low-rank structure of neural network Hessians.
problem Understanding the structure of Hessians in neural networks.
method Proposes a decoupling conjecture to decompose layer-wise Hessians into Kronecker products of smaller matrices.
result Proves the structure of top eigenspaces in 2-layer networks and shows high overlap in top eigenvectors across different models.
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.
Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.
problem Global topological constraints and structural properties of compact Hessian manifolds.
method Novel fibration and splitting theorems, Chern's conjecture, Hitchin systems, Cheng-Yau solution.
result Topological classification of complete Hessian surfaces and closed orientable Hessian 3-manifolds.
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 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.
Bounds on Hessian of heat equation coupled with Ricci flow.
problem Estimating the Hessian of a solution to the conjugate heat equation coupled with Ricci flow.
method Obtained upper bounds for the Hessian.
result Local and global upper bounds for the Hessian of a positive solution.
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.
New perspective on CNNs using Hessian maps reveals their structure.
problem Understanding the nature of Convolutional Neural Networks (CNNs).
method Developed a framework using Toeplitz representation of CNNs to reveal Hessian structure and prove rank bounds.
result Proved that the Hessian rank of CNNs grows as the square root of the number of parameters.
Paper studies Hessian quotient equations in warped product manifolds.
problem Analyzing Hessian quotient equations in warped product manifolds.
method Using standard degree theory and a priori estimates.
result Existence of star-shaped compact hypersurface solutions.