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. New proof for stability estimates in complex equations without pluripotential theory.
problem Stability estimates for complex Monge-Ampère and Hessian equations.
method New proof using general degenerations of background metrics.
result Uniform stability estimates for both equations under various degenerations.
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.
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. The note provides uniform estimates for complex Hessian equations on compact Hermitian manifolds.
problem Uniform estimates for solutions to degenerate complex Hessian equations on compact Hermitian manifolds.
method The approach relies on corresponding a priori estimates for Monge-Ampère equations.
result Extension and short alternative proof of results for complex Hessian equations.
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.
Let (X,ω) be a compact Kähler manifold of dimension n and fix m∈N such that 1≤m≤n. We prove that any (ω,m)-sh function can be approximated from above by smooth (ω,m)-sh functions. A potential theory for the complex Hessian equation is also developed which generalizes the classical pluri…
Let (X,ω) be an n-dimensional compact Kähler manifold. We study degenerate complex Hessian equations of the form (ω+ddcφ)m∧ωn−m=F(x,φ)ωn. Under some natural conditions on F, this equation has a unique continuous solution. When (X,ω) is rational homogeneous we further show that the solu…
Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to understand Schauder type estimates fashioning particular borderline scenarios. In such c…
Stochastic gradient descent (SGD) forms the core optimization method for deep neural networks. While some theoretical progress has been made, it still remains unclear why SGD leads the learning dynamics in overparameterized networks to solutions that generalize well. Here we show that for overparameterized networks wit…
We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the C1,1 boundary data, we establish the interior C1,1-regularity of the unique (admissible) solution, which is o…
Introduces a new geometric structure for statistical manifolds with degenerate metrics.
problem Degenerate metrics in statistical manifolds affect geometric structures and applications.
method Introduces quasi-Codazzi structure for degenerate metrics and coherent tangent bundles.
result Generalizes geometric structures and relations for statistical models with degenerate metrics.
Derives estimates for geometric elliptic equations on complex manifolds.
problem Estimating solutions of geometric elliptic equations on complex manifolds.
method Derives a priori real Hessian estimates independent of the right-hand side.
result Establishes optimal C1,1 regularity of geometric envelopes. 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 studies critical points in overparameterized neural networks, identifying a star locus and degenerate critical points.
problem Understanding the geometry of loss functions in overparameterized neural networks.
method Identifying and analyzing components of the critical locus of the loss function L for overparameterized feedforward neural networks of depth ℓ≥4. result For very wide networks, all critical points are degenerate, and lower bounds on the number of zero eigenvalues of the Hessian are given.
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.
The paper finds geodesics in Kähler potentials with no degeneration.
problem Finding geodesics in Kähler potentials without degeneration.
method Establishing a lower bound estimate for eigenvalues of complex Hessian.
result Geodesics can connect close points in Kähler potentials without degeneration.
Let U⊂An be an open subset of real affine space. We consider functions F:U→R with non-degenerate Hessian such that the first or the third derivative of F is parallel with respect to the Levi-Civita connection defined by the Hessian metric F". In the former case the solutions are gi…
Hessian metrics on a sphere with specific properties.
problem Existence and properties of Hessian metrics on a sphere with singularities.
method Construction of pseudo-metric tensor with distribution coefficients.
result Existence of Hessian metrics on a sphere with specific properties.
There are two parts of this paper. First, we discovered an explicit formula for the complex Hessian of the weighted log-Bergman kernel on a parallelogram domain, and utilised this formula to give a new proof about the strict convexity of the Mabuchi functional along a smooth geodesic. Second, when a C^{1,1}-geodesic co…
A new geometric structure for singular models is introduced.
problem Degenerate metrics in dually flat structures.
method Introducing quasi-Hessian manifolds with degenerate metrics and symmetric cubic tensors.
result Established Amari-Nagaoka's extended Pythagorean and projection theorems for singular models.
New geometric structures derived from Hessian metrics and tensors.
problem Constructing geometric structures from Hessian metrics and tensors.
method Introduced a family of Hessian metrics and constructed a (1,1)-tensor field. result Derived golden and metallic structures from the tensor field.
Proves smooth solutions for generalised Monge-Ampère equations on projective manifolds.
problem Existence of smooth solutions for generalised Monge-Ampère equations on projective manifolds.
method Intersection numbers and degenerate concentration of mass result.
result Proves existence of smooth solutions for generalised Monge-Ampère equations on projective manifolds.
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.
We establish an unexpected relation among the Weil-Petersson metric, the generalized Hodge metrics and the BCOV torsion. Using this relation, we prove that certain kind of moduli spaces of polarized Calabi-Yau manifolds do not admit complete subvarieties. That is, there is no complete family for certain class of polari…
The paper explores the geometric structure of cost functions in multiple dimensions.
problem Understanding the geometric properties of cost functions in multidimensional settings.
method Analyzes the Hessian metric and geodesics in logarithmic and original coordinates.
result The geometry is one-dimensional in logarithmic coordinates but effectively (n−1)-dimensional in original coordinates. We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and examples are provided. This continues the investigation begun in [ABCDS] where the met…
Paper finds local normal forms for wavefronts in flat coordinates.
problem Understanding local diffeomorphic types of wavefronts.
method Using connections and the metric, criteria for wavefront types are derived in affine flat coordinates.
result Local normal forms of e/m-wavefronts in affine flat coordinates are derived. BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M x N of a quaternionic-Kaehler manifold M of negative scalar curvature and a very special real manifold N of dimension n >=0. Such gradient flows are generated by the `energy function' f = P^2, where P is a (bundle-valued) …
Study on polyhedra rigidity, finding non-existence of flexible weakly convex decomposable polyhedra.
problem Proving all decomposable polyhedra with vertices in convex position are infinitesimally rigid.
method Constructing explicit families of polyhedra, using the Hessian of the discrete Hilbert-Einstein functional, and searching for eigenvalues of the Hessian with Mathematica.
result Experimental evidence suggests no flexible, weakly convex and decomposable polyhedra exist.
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…
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.
The landscape of empirical risk has been widely studied in a series of machine learning problems, including low-rank matrix factorization, matrix sensing, matrix completion, and phase retrieval. In this work, we focus on the situation where the corresponding population risk is a degenerate non-convex loss function, nam…
We propose a fast second-order method that can be used as a drop-in replacement for current deep learning solvers. Compared to stochastic gradient descent (SGD), it only requires two additional forward-mode automatic differentiation operations per iteration, which has a computational cost comparable to two standard for…
New superintegrable systems derived from Frobenius structures.
problem Constructing second-order superintegrable systems.
method Using conification and direct product construction, applying to semi-simple and nilpotent algebras.
result Explicitly constructed second-order superintegrable systems in three dimensions.
Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing th…
This article is devoted to the variational study of two functions defined over some Teichmueller spaces of hyperbolic surfaces. One is the systole of geodesic loops based at some fixed point, and the other one is the systole of arcs.\par For each of them we determine all the critical points. It appears that the systole…
For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of second-order jet space) and are defined by non-degenerate scalar second-order-only (Hessian) PDEs in…
NGD models have higher effective dimension than SGD models.
problem Measuring model complexity accurately.
method Comparison of NGD and SGD models using effective dimension measures.
result NGD models have a higher effective dimension than SGD models.
The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes …
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.
Study neural architectures on learned latent graphs using Schrödinger dynamics.
problem Understanding neural architectures on learned latent graphs.
method Optimizes over stratified moduli space of weighted graphs with Kähler-Hessian metric.
result Multilayer stationary networks are equivalent to global stationary problems on supra-graphs.
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.