Noise injection regularizes Hessian, improving neural network training and generalization.
problem Regularizing over-parameterized neural networks with nonconvex and nonlinear geometry.
method Injecting isotropic Gaussian noise into weight matrices and designing a two-point estimate of the Hessian penalty.
result Effective regularization of Hessian improves generalization, achieving up to 2.4% test accuracy increase.
Zeroth-order methods favor flat minima in machine learning.
problem Finding solutions with small Hessian trace in optimization.
method Zeroth-order optimization with two-point estimator.
result Zeroth-order optimization converges to flat minima.
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.
Study the geometry of a Lie group using Hessian and flat affine structures.
problem Understanding the geometry of a specific Lie group.
method Examined using bi-invariant Hessian metric and flat affine structure, focusing on curvatures, causal structure, and developed map.
result Determined curvatures, tidal force, and Jacobi vector fields of the Hessian metric.
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…
Study geometric properties of loss functions to understand neural network performance.
problem Understanding the geometric properties of high-dimensional loss functions to improve neural network performance.
method Combine concepts from high-dimensional probability and differential geometry to study curvature properties in lower-dimensional loss representations.
result Mean curvature in the original loss space determines if saddle points appear as minima, maxima, or flat regions.
Better Hessian approximations improve influence function attributions in deep learning.
problem Influence functions are difficult to compute due to ill-conditioned Hessians, leading to poor data attribution performance.
method Investigated the impact of Hessian approximation quality on influence-function attributions in a controlled setting.
result Better Hessian approximations consistently yield better influence score quality.
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.
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.
New findings challenge the use of flatness measures in neural networks.
problem The validity of flatness measures in assessing generalization in neural networks.
method Analysis of Hessian-based flatness norms and their relation to generalization.
result Solutions with large weights and low loss are often sharper than expected, contradicting flatness measures.
Transformers use a unique Hessian structure that differs from classical networks, affecting optimization.
problem Understanding the unique optimization landscape of Transformers.
method Theoretical Hessian analysis of a single self-attention layer in Transformers.
result Transformers have a highly non-linear Hessian structure, distinguishing them from classical networks.
HTE improves PINNs for high-dimensional, high-order PDEs by reducing computational cost and memory usage.
problem Challenges in solving high-dimensional, high-order PDEs with PINNs due to computational cost and memory constraints.
method Introduces Hutchinson Trace Estimation (HTE) to transform Hessian matrix calculations into Hessian vector products (HVP), reducing computational cost and memory usage.
result HTE significantly reduces memory consumption and computational cost, enabling faster and more efficient solution of high-dimensional and high-order PDEs.
Studied how SGD's stability regularization affects generalization in neural networks.
problem Understanding why SGD often generalizes better than GD in neural networks.
method Analyzed stability of SGD and GD through Frobenius norm and trace of Hessian, and compared their generalization properties.
result Stable minima of SGD generalize well, while GD's stability-induced regularization is too weak.
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.
GL-LowPopArt improves minimax-optimal estimation for trace regression.
problem Minimizing estimation error in generalized low-rank trace regression.
method Two-stage approach: nuclear norm regularization followed by matrix Catoni estimation.
result Achieves instance-wise optimal error bounds up to condition number.
SGD without replacement decouples into curvature-following and flatness-regularizing steps.
problem Theoretical analysis of SGD without replacement for large-scale neural networks.
method Analysis of SGD without replacement in a realistic regime, considering high curvature and flatness.
result Optimizing with SGD without replacement is locally equivalent to an additional regularizer step.
Paper proposes a new flatness measure for neural networks to improve generalization.
problem Generalization in deep learning models, especially with overparameterization.
method Soft rank measure of the Hessian to assess flatness and generalization.
result Soft rank flatness measure accurately estimates generalization gaps for various models.
Studying SGD on deep neural networks using diffusion maps.
problem Understanding why SGD performs well in deep learning.
method Data-driven approach using diffusion maps to analyze SGD dynamics.
result SGD dynamics may mainly live on a low-dimensional manifold in high-dimensional parameter space.
Flat minima lead to better generalization in low-rank matrix recovery models.
problem Understanding why flat minima generalize well in overparameterized models.
method Analysis of overparameterized matrix and bilinear sensing, robust PCA, covariance matrix estimation, and neural networks with quadratic activation functions.
result Flat minima, measured by the trace of the Hessian, exactly recover the ground truth in low-rank matrix recovery models under standard statistical assumptions.
In distributed optimization and distributed numerical linear algebra, we often encounter an inversion bias: if we want to compute a quantity that depends on the inverse of a sum of distributed matrices, then the sum of the inverses does not equal the inverse of the sum. An example of this occurs in distributed Newton's…
The paper studies the loss landscape of regularized deep matrix factorization, revealing unique and sharp minimizers.
problem Understanding the loss landscape and minimizers of regularized deep matrix factorization problems.
method Theoretical analysis of ℓ2-regularized deep matrix factorization/deep linear network training problems with squared-error loss. result The unique end-to-end minimizer exists for all target matrices except for a set of Lebesgue measure zero.
We establish a classification of cubic minimal cones in case of the so-called radial eigencubics. Our principal result states that any radial eigencubic is either a member of the infinite family of eigencubics of Clifford type, or belongs to one of 18 exceptional families. We prove that at least 12 of the 18 families a…
Global convergence of an online (stochastic) limited memory version of the Broyden-Fletcher- Goldfarb-Shanno (BFGS) quasi-Newton method for solving optimization problems with stochastic objectives that arise in large scale machine learning is established. Lower and upper bounds on the Hessian eigenvalues of the sample …
How can we explain the predictions of a black-box model? In this paper, we use influence functions -- a classic technique from robust statistics -- to trace a model's prediction through the learning algorithm and back to its training data, thereby identifying training points most responsible for a given prediction. To …
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.
The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certa…
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 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.
We propose a new algorithm for finite sum optimization which we call the curvature-aided incremental aggregated gradient (CIAG) method. Motivated by the problem of training a classifier for a d-dimensional problem, where the number of training data is m and m≫d≫1, the CIAG method seeks to accelerate increme…
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.
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.
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.
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.