A selfsimiar manifold is a Riemannian manifold endowed with a homothetic vector field . 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…
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48 results for “Local Hessian”
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.
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.
Upper Bounds for Hessian Matrices of Positive Solutions to Heat Equations on Kähler Manifoldsmath.DG
The study sets limits on heat equation solutions' Hessians on curved spaces.
problem Bounding Hessians of positive solutions to heat equations on Kähler manifolds.
method Global and local upper bounds for Hessian matrices under curvature constraints.
result Improved bounds on Hessians for Riemannian manifolds with lower sectional curvature.
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.
A new method simplifies HLLE for better robustness.
problem Improving robustness of Hessian locally linear embedding.
method Replacing Hessian with arbitrary weights and modifying manifold dimension.
result Achieved a new LLE-type method called tangential LLE.
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 -wavefronts in affine flat coordinates are derived.
New averaging technique speeds up Newton method convergence.
problem Superlinear convergence of stochastic Newton methods with noisy Hessians.
method Hessian averaging to reduce noise and maintain superlinear convergence.
result Hessian averaging achieves superlinear convergence with a non-asymptotic rate.
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.
New methods optimize functions faster with less gradient accuracy needed.
problem Optimizing complex functions with limited gradient accuracy.
method Hessian averaging and adaptive gradient sampling methods.
result Improved convergence rates for various function types.
Paper improves a method for fast global and local convergence in optimization.
problem Slow global convergence in optimization methods with noisy Hessian estimates.
method Stochastic Newton Proximal Extragradient method using HPE framework.
result Faster global linear rate and superlinear convergence in fewer iterations.
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 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.
While it has not yet been proven, empirical evidence suggests that model generalization is related to local properties of the optima which can be described via the Hessian. We connect model generalization with the local property of a solution under the PAC-Bayes paradigm. In particular, we prove that model generalizati…
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.
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 understanding of the dynamics of the velocity gradients in turbulent flows is critical to understanding various non-linear turbulent processes. The pressure-Hessian and the viscous-Laplacian govern the evolution of the velocity-gradients and are known to be non-local in nature. Over the years, several simplified dy…
Local constancy of index for certain gradient mappings proved.
problem Proving the local constancy of the index for specific gradient mappings.
method Using a more general theorem for quasiregular gradient mappings, deducing the result from the Hessian's properties.
result The index is locally constant for functions with uniformly positive determinant Hessian almost everywhere.
Smooth surfaces can always be locally described by Hessians.
problem Locally describing nondegenerate surface metrics as Hessians.
method Analyzing smooth surfaces and their metrics in local coordinates.
result Smooth surfaces can always be locally described by Hessians.
We target the problem of finding a local minimum in non-convex finite-sum minimization. Towards this goal, we first prove that the trust region method with inexact gradient and Hessian estimation can achieve a convergence rate of order as long as those differential estimations are sufficientl…
We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolves under the Ricci flow. These upper bounds supplement the well-known global lower bound.
The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.
problem Estimating solutions to nonlinear weighted parabolic equations.
method Derives Li-Yau and Hamilton type gradient estimates, and Hessian estimates.
result New gradient and Hessian estimates for positive solutions of nonlinear parabolic equations.
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.
We propose a sample efficient stochastic variance-reduced cubic regularization (Lite-SVRC) algorithm for finding the local minimum efficiently in nonconvex optimization. The proposed algorithm achieves a lower sample complexity of Hessian matrix computation than existing cubic regularization based methods. At the heart…
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
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.
The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundarymath.DG
We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a …
Let be an open subset of real affine space. We consider functions with non-degenerate Hessian such that the first or the third derivative of is parallel with respect to the Levi-Civita connection defined by the Hessian metric . In the former case the solutions are gi…
HA-SME models SGD dynamics with Hessian info for better escaping behaviors.
problem Capturing the escaping behaviors of SGD from stationary points.
method HA-SME, a novel SDE with Hessian info in drift and diffusion.
result HA-SME achieves best approximation error and recovers SGD dynamics for quadratics.
We analyze the Hessian spectra of large models up to 100B parameters.
problem Accurate Hessian spectra of large foundation models are difficult to obtain.
method We use shard-local finite-difference Hessian vector products and stochastic Lanczos quadrature.
result We produce the first large-scale spectral density estimates of foundation models.
The paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.
problem Discrete construction of Hessian and divdiv complexes on triangulations.
method Construction of discrete Hessian and divdiv complexes using finite elements and Dirac measures on triangulations.
result The cohomology of the constructed complexes is isomorphic to the continuous de Rham cohomology.
The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the section u can be regarded as a metric potential analogous to the local potential …
Debiasing Distributed Second Order Optimization with Surrogate Sketching and Scaled Regularizationcs.LG
New technique debiases distributed optimization, improving convergence rate.
problem Bias in local estimates limits effectiveness of distributed second order optimization.
method Surrogate sketching and scaled regularization to eliminate bias.
result The debiased local estimates lead to faster convergence in distributed optimization.
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.
Study shows polynomial-width neural networks can closely approximate infinite-width networks in polynomial time.
problem Approximating dynamics of polynomial-width neural networks with infinite-width networks.
method Bounding approximation gap through a differential equation governed by mean-field dynamics, considering local Hessian.
result Polynomially many neurons are sufficient to closely approximate mean-field dynamics.
We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from estimates. Also, the method is flexible and can be applied to a large class of equations.
We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this …
The study explores Hesse manifolds and their symmetries in multifield cosmological models.
problem Understanding symmetries in multifield cosmological models.
method Analyzes Hesse functions and their properties on Riemannian manifolds.
result Complete Hesse manifolds are characterized by their index and are hyperbolic.
The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.
problem Generalizing differential geometric structures to algebroids.
method Introducing statistical, conjugate connection, and Hessian structures on anti-commutable pre-Leibniz algebroids.
result Statistical and conjugate connection structures are equivalent for admissible connections.
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 .
result The difference is determined by the logarithmic differential of a Hessian potential function.
We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.
This research analyzes and accelerates score-based diffusion models using discretization and Hessian information.
problem Theoretical foundations and convergence analysis of score-based diffusion models.
method Investigation of various discretization schemes, including Euler, exponential integrators, and midpoint randomization. Proposal of an accelerated sampler based on local linearization method.
result Hessian-based approach achieves faster convergence rates of order $\widetilde{\mathcal{O}}\left(\frac{1}{\varepsilon}
ight)$, significantly improving upon vanilla diffusion models.
The paper studies Minkowski norms and Hessian isometries induced by isoparametric foliations on spheres.
problem Understanding Minkowski norms and Hessian isometries induced by isoparametric foliations.
method Constructing Minkowski norms using spherical coordinates, studying Hessian isometries using spherical local frames, and proving properties of these isometries.
result Proves the existence and properties of Hessian isometries induced by isoparametric foliations on spheres.
Study uncovers complex critical points in tensor decomposition.
problem Nonconvex optimization of symmetric tensor decomposition.
method Utilized symmetry to construct critical points and analyze Hessian.
result Obtained precise analytic estimates on objective function and Hessian.
Random Matrix Theory explains loss surface Hessians in neural networks.
problem Understanding the loss surfaces of neural networks.
method Investigation of local spectral statistics of neural network Hessians.
result Excellent agreement with Gaussian Orthogonal Ensemble statistics.
Gradient descent forces neural network eigenvalues to a specific threshold.
problem Understanding why gradient descent drives eigenvalues to a specific threshold.
method Introduced edge coupling, a functional on consecutive iterate pairs, to explain the trajectory towards the eigenvalue threshold.
result Gradient descent forces the Hessian eigenvalue to the threshold from arbitrary initialization.
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…
We propose a modular extension of backpropagation for the computation of block-diagonal approximations to various curvature matrices of the training objective (in particular, the Hessian, generalized Gauss-Newton, and positive-curvature Hessian). The approach reduces the otherwise tedious manual derivation of these mat…