Kunneth formula derived for flat affine manifolds and applied to Hessian metrics.
problem Cohomology of flat affine manifolds and Hessian metrics.
method Proved Künneth formula for Dolbeault--Koszul cohomology of flat affine manifolds and applied to Hessian metrics.
result Kunneth formula for Hessian metrics on products and manifolds with hyperbolic manifolds.
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.
Proves triviality and nonexistence of gradient Ricci solitons as warped metrics.
problem Proving triviality and nonexistence of gradient Ricci solitons as warped metrics.
method Proved through the construction of gradient Ricci solitons as warped products and studying Ricci-Hessian type manifolds.
result Gradient Ricci solitons are trivial and non-existent as warped metrics.
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 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.
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.
Study proves structure results for homogeneous spaces supporting specific equations.
problem Proving structure results for homogeneous spaces supporting specific equations.
method Analyzing homogeneous spaces with non-constant solutions to two general classes of equations involving the Hessian and an invariant 2-tensor.
result Generalizes rigidity results for gradient Ricci solitons and warped product Einstein metrics.
In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. We obtain a uniqueness result for prescribing the Ricci curvatur…
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.
The paper studies convexity of products of squared Euclidean distances.
problem Convexity of products of squared Euclidean distances.
method Proved a convexity principle and applied it to products of squared distances, computed Hessian-positive regions and exact convexity levels.
result Computed exact convexity and quasiconvexity truncation levels for the two-centre model.
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.
Derives formulas from Green function Hessian assumption.
problem Deriving formulas from Green function Hessian assumption.
method Assumption on Hessian of Green function leads to monotonicity formulas.
result Explicit examples of manifolds satisfying assumption.
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…
HessFormer enables distributed Hessian computation for large models.
problem Limited Hessian computation for large deep learning models.
method Distributed stochastic Lanczos quadrature algorithm.
result Investigated Hessian spectral density of a 70 billion parameter model.
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.
The paper studies special contact metric manifolds and their properties.
problem Investigating properties of contact metric manifolds with a specific equation.
method Analyzing K-contact and (κ,μ)-contact manifolds with a smooth function f satisfying a given equation. result Complete and simply connected K-contact manifolds admitting such a function are isometric to the unit sphere. A new optimisation method efficiently scales Hessian-vector products for neural networks.
problem Challenges in applying second-order quasi-Newton methods due to large Hessian and non-convexity.
method Proposes an optimisation algorithm that asymptotically uses the exact inverse Hessian with modified eigenvalues.
result Demonstrates scalability and comparable performance to other optimisation methods in neural networks.
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. 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…
ASTRA improves TDA by more accurately approximating iHVP.
problem Improving insights into training data attribution.
method ASTRA uses EKFAC-preconditioner on Neumann series iterations to accurately approximate iHVP.
result Improving iHVP approximation significantly improves TDA performance.
Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have received increasing attention due to its improved gradient/Hessian complexities (i.e., number of queries to stochastic gradient/Hessian oracles) to find local minima for nonconvex finite-sum optimization. However, it is unclear whether existing SVR…
New SGD algorithm finds critical points faster with second-order corrections.
problem Finding critical points in non-convex optimization efficiently.
method Uses Hessian-vector products to correct momentum bias in SGD.
result Finds ε-critical points in O(ε−3) time. The Hesse-Koszul flow converges to the Hesse-Einstein metric on compact Hessian manifolds.
problem Existence and convergence of Hesse-Einstein metrics on compact Hessian manifolds.
method Study of the Hesse-Koszul flow and its convergence properties.
result The flow converges to the unique Hesse-Einstein metric under certain conditions.
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.
The study finds starshaped compact hypersurfaces in warped products with curvature estimates.
problem Finding starshaped compact hypersurfaces in warped product manifolds.
method Deriving global curvature estimates and interior second order a priori estimates for solutions to associated equations.
result Existence of starshaped compact hypersurfaces in warped product manifolds.
No non-product Hessian rank 1 affine homogeneous hypersurfaces exist in dimensions 5 and above.
problem Identifying non-product Hessian rank 1 affine homogeneous hypersurfaces in higher dimensions.
method Developed a normal form for hypersurfaces under the affine group, up to order ≤ n+5, in any dimension n ≥ 2.
result Non-existence of non-product Hessian rank 1 affine homogeneous hypersurfaces in dimensions 5 and above.
We study the Hessian geometry of toric Gibbons-Hawking metrics and their phase change phenomena via the images of their moment maps.
Study reveals flatness of Hessian metrics with non-negative Ricci curvature on foliation leaves.
problem Rigidity of Ricci curvature on Hessian manifold leaves.
method Analysis of Ricci curvature properties of Hessian metrics on foliation leaves.
result Non-negative Ricci curvature on a single leaf forces the Hessian metric to be flat and yields bounds on the first Betti number.
Shampoo optimizes preconditioners for faster convergence in machine learning.
problem Improving convergence speed in machine learning optimization.
method Explicit connection between Shampoo's Kronecker product approximation and optimal matrix approximations.
result The square of Shampoo's approximation is equivalent to a single power iteration step for optimal Kronecker product approximation.
We study the Hessian geometry of toric multi-Taub-NUT metrics and their phase change phenomena via the images of their moment maps. This generalizes an earlier paper on toric Gibbons-Hawking metrics.
New method for efficient sketching of gradients and Hessians.
problem Memory constraints in training machine learning models.
method A novel framework for scalable gradient and HVP sketching tailored for modern hardware.
result Theoretical guarantees and practical applications in training data attribution and Hessian spectrum analysis.
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.
We extend the correspondence between Hessian and Kähler metrics and curvatures to Lagrange spaces.
Study of pseudo-Riemannian metrics related to Monge-Ampère structures.
problem Properties of pseudo-Riemannian metrics and Hessian structures.
method Analysis of Monge-Ampère structures and Ricci flat solutions.
result Description of a family of Ricci flat solutions parametrized by six coefficients.
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.
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 introduces the Metric-Free Natural Gradient (MFNG) algorithm for training Boltzmann Machines. Similar in spirit to the Hessian-Free method of Martens [8], our algorithm belongs to the family of truncated Newton methods and exploits an efficient matrix-vector product to avoid explicitely storing the natural g…
The paper studies hybrid connections on Hessian manifolds and their properties.
problem Investigating hybrid connections on Hessian manifolds.
method Defining and analyzing hybrid connections as incompressible affine connections projective to a flat connection D. result The difference abla−D is determined by the logarithmic differential of a Hessian potential function. 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.
Paper develops a splitting principle for RCD spaces, extending manifold properties.
problem Understanding splitting properties in RCD spaces.
method General analytic splitting principle for RCD spaces.
result Spaces with suitable functions have splitting properties.
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.
Study classifies special metrics on specific surfaces.
problem Classifying extremal Kähler metrics with singularities.
method Analyzes Hessian of the Curvature of the Metric on K-surfaces.
result Identifies non-CSC HCMU metrics on S{α}2 and S{α,β}2. 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.
Locally conformally Hessian manifolds are dense in radiant ones of rank 1.
problem Characterizing locally conformally Hessian manifolds and their properties.
method Analyzing quotient spaces of Hessian manifolds and using statistical manifold theory.
result The set of radiant l.c.H. metrics of rank 1 is dense in all radiant l.c.H. metrics.
Study of manifolds with flat connections and diagonal metrics leading to vanishing Euler characteristic.
problem Understanding the geometry and topology of affine-orthogonal manifolds.
method Deformation of flat connections into Levi-Civita connections and analysis of Euler characteristic.
result Deformations force the Euler characteristic to vanish, supporting Chern's conjecture.
WoodFisher improves neural network compression efficiency and accuracy.
problem Efficiently estimating inverse Hessian for neural network optimization.
method WoodFisher: a method to compute a faithful and efficient estimate of the inverse Hessian.
result WoodFisher significantly outperforms state-of-the-art methods for pruning neural networks.
Abstract: Characterizes special Kähler manifolds with specific properties.
problem Characterizing Kähler manifolds with special properties.
method Analyzes properties of functions and gradients on manifolds.
result Characterizes manifolds supporting certain functions and gradients.
The study classifies Hessian rank 1 hypersurfaces in dimensions 2, 3, and 4.
problem Classifying Hessian rank 1 affinely homogeneous hypersurfaces in specific dimensions.
method Power Series Method of Equivalence, infinitesimal calculations.
result Identified all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces in dimensions 2, 3, and 4.