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. We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in Lp for some p>1. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on $\mathbb…
Author presents the second variational formula for statistical biharmonic maps.
problem Developing a formula for statistical biharmonic maps.
method Introduced the second variational formula for the statistical bi-energy functional.
result The second variational formula can be represented using Hessian curvature in Hessian manifolds.
Optimizes quadratic bandits with tight Hessian-dependent sample complexity bounds.
problem Understanding optimal sample complexity for quadratic functions.
method Introduces energy allocation and optimal energy spectrum to prove tight lower bounds. Solves for Hessian-independent optimal algorithm.
result Proves optimal Hessian-dependent sample complexities and existence of a universally optimal algorithm.
We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in Lp for p>34. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic ma…
The paper introduces a new system of equations for Hessian-cscK metrics.
problem Finding constant scalar curvature Kähler metrics.
method Proposes a coupled system of complex Hessian equations and shows it can be variational.
result Proves a C0-estimate for the system that depends on entropy. The most fruitful approach to studying low energy soliton dynamics in field theories of Bogomol'nyi type is the geodesic approximation of Manton. In the case of vortices and monopoles, Stuart has obtained rigorous estimates of the errors in this approximation, and hence proved that it is valid in the low speed regime. …
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.
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…
The Bethe free energy approximation is reliable when convex on a submanifold, the 'Bethe box'.
problem Accuracy of the Bethe free energy approximation in probabilistic inference.
method Analysis of convexity and verification conditions based on the Bethe Hessian matrix.
result The Bethe approximation is mostly accurate if it is convex on a submanifold, the 'Bethe box'.
New quasimetric spaces improve stability in complex Hessian equations.
problem Improving stability results for complex Hessian equations.
method Constructing a family of quasimetric spaces in generalized potential theory.
result Convergence of quasimetric spaces leads to improved stability results.
Solves complex Hessian equations in unstable cases, proving unique canonical solutions with singularities.
problem Existence of smooth solutions to complex Hessian equations in unstable cases.
method Parabolic flows and moment-map energy functionals, focusing on J-equation and deformed Hermitian Yang-Mills equation.
result Proves existence of unique canonical solutions with singularities on Kahler surfaces.
Study on heat kernel asymptotics and path integrals on Riemannian manifolds.
problem Investigating the short-time expansion of heat kernel on compact Riemannian manifolds.
method Formally expressing the heat kernel as a path integral and using Laplace's method.
result The lowest order term of the heat kernel's short-time expansion is given by the Fredholm determinant of the Hessian of the energy functional.
In this brief survey, we will remark the interaction among the Hessian tensor on a semi-Riemannian manifold and some of the several questions in Lorentzian (and also in semi-Riemannian) geometry where this 2-covariant tensor is involved. In particular, we deal with the characterization of Killing vector fields and the …
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.
Calculates metrics and geodesics for symplectic forms.
problem Computing metrics and geodesics for symplectic forms.
method Computed the Levi-Civita connection, described geodesics, and computed the covariant Hessian.
result Formula for the covariant Hessian of an energy functional.
We study the small time asymptotics of the gradient and Hessian of the logarithm of the heat kernel at the cut locus, giving, in principle, complete expansions for both quantities. We relate the leading terms of the expansions to the structure of the cut locus, especially to conjugacy, and we provide a probabilistic in…
The paper proves strict convexity of the Mabuchi functional for geodesics connecting energy minimizers.
problem Proving strict convexity of the Mabuchi functional for geodesics.
method Explicit formula for the complex Hessian of the weighted log-Bergman kernel, and proof by showing geodesics must be non-degenerate and smooth.
result Strict convexity of the Mabuchi functional along geodesics connecting energy minimizers.
Neural networks' energy landscape is surprisingly flat, suggesting minimal structural changes between minima.
problem Understanding the structure of neural network energy landscapes.
method Constructing continuous paths between minima of recent neural network architectures on CIFAR10 and CIFAR100.
result Paths between minima are essentially flat in both training and test landscapes, implying minimal structural changes.
Let M be a closed Riemann surface, N a Riemannian manifold of Hermitian non-positive curvature, f:M→N a continuous map, and E the function on the Teichmüller space of M that assigns to a complex structure on M the energy of the harmonic map homotopic to f. We show that E is a plurisubharmonic functio…
New method accelerates neural network training by focusing on flat directions.
problem Improving neural network training speed and stability.
method Bulk-SGD, interpolated gradient methods.
result Updates along the Dominant subspace can accelerate convergence but compromise stability.
The formal structure of geometrical thermodynamics is reviewed with particular emphasis on the geometry of equilibria submanifolds. On these submanifolds thermodynamic metrics are defined as the Hessian of thermodynamic potentials. Links between geometry and thermodynamics are explored for single and multiple component…
The paper quantizes the energy of Yang-Mills connections on stationary bundles.
problem Quantifying the energy of Yang-Mills connections on stationary bundles.
method Analyzes the convergence of sequences of connections and uses bubble tree decomposition.
result Explicit computation of energy density from blow-ups of Yang-Mills connections.
The energy function associated to harmonic maps between surfaces is convex at critical points.
problem Proving convexity of the energy function for harmonic maps between surfaces.
method Analyzing the energy function on Teichmüller space and proving convexity at critical points.
result The energy function is convex at critical points and strictly convex under certain conditions.
The paper introduces a new energy density function and proves Liouville type theorems for various maps.
problem Proving Liouville type theorems for holomorphic, harmonic, and pluri-harmonic maps.
method Introducing a new energy density function and deriving Hessian estimates.
result No non-constant holomorphic map exists between certain Hermitian manifolds with specific curvature conditions.
Model explains surprising properties of neural network training landscapes.
problem Understanding the geometry of neural network loss landscapes.
method Developed a simple theoretical model of gradients and Hessians.
result Unified model accounts for 4 surprising properties of neural loss landscapes.
In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. T…
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).
We study the deformations of twisted harmonic maps f with respect to the representation ρ. After constructing a continuous "universal" twisted harmonic map, we give a construction of every first order deformation of f in terms of Hodge theory; we apply this result to the moduli space of reductive representations …
The paper examines conditions for the equator map to be minimizing or unstable for higher order energy functionals.
problem Conditions for the equator map to be minimizing or unstable for extrinsic k-energy functionals.
method Analyzes the extrinsic k-energy functional and the equator map to establish conditions for minimization or instability.
result Establishes necessary and sufficient conditions for the equator map to be minimizing or unstable for extrinsic k-energy functionals.
In this work we study the intrinsic geometry of the space of Kahler metrics under various Riemannian metrics. The first part is on the Dirichlet metric. We motivate its study, we compute its curvature, and we make links with the Calabi metric, the K-energy, the degenerate complex Hessian equation. The second part is on…
Entropy-SGD improves deep learning by favoring flat minima in the energy landscape.
problem Training deep neural networks to avoid poorly-generalizing sharp minima.
method Constructs a local-entropy-based objective function to bias SGD towards flat regions.
result Entropy-SGD leads to improved generalization over SGD, as shown by experiments and theoretical underpinnings.
Study energy landscapes in glass models, focusing on Gaussian and spiked-tensor functions.
problem Characterize statistical properties and phase transitions of high-dimensional energy landscapes.
method Developed a Kac-Rice method framework to compute landscape complexity and analyze phase transitions rigorously.
result Characterized the ruggedness and arrangements of local minima in energy landscapes.
Active learning reduces SP calculations by 90%.
problem Efficiently calculating saddle points in energy functions.
method Active learning framework with GPR and GAD.
result Significant reduction in the number of expensive evaluations.
Researchers solve a complex equation to embed graphs with negative curvature.
problem Embedding graphs in Rn+1 with negative Gauss curvature. method Solving a fully nonlinear Monge-Ampère equation using energy estimates and Nash-Moser iteration.
result Local solvability of the fully nonlinear equation for negative curvature.
New algorithms improve tensor PCA performance using Kikuchi Hessian.
problem Tensor PCA problem.
method Hierarchy of spectral methods based on Kikuchi Hessian.
result Polynomial-time algorithm matching SOS performance.
Gaussian process regression speeds up nudged elastic band calculations for transitions.
problem Reducing computational effort for calculating minimum energy paths in thermalized systems.
method Approximate energy surface generation and refinement using Gaussian process regression.
result The number of energy and force evaluations can be reduced by an order of magnitude.
New method estimates Nishimori temperature for node classification in weighted graphs.
problem Estimating Nishimori temperature for Bayesian inference.
method Spectral method using eigenvalues of Bethe Hessian matrix.
result Spectral method outperforms existing approaches in node classification.
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.
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.
A new method for estimating complex models and high-dimensional data.
problem Difficulty in computing Hessian of log-density functions for complex models and high-dimensional data.
method Sliced score matching, which projects scores onto random vectors before comparison.
result Sliced score matching can learn deep energy-based models and produce accurate score estimates.
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 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.