arXiv research

The article derives gradient estimations for semilinear equations on geometric flows.

GeoAdaLer enhances geometric understanding of Adam for stochastic optimization.

Extends gradient estimates for heat equation under Finsler geometric flows.

The paper estimates gradients for a weighted parabolic equation under geometric flow.

Extends geometric structures to manifolds with new operators.

In general, gradient estimates are very important and necessary for deriving convergence results in different geometric flows, and most of them are obtained by analytic methods. In this paper, we will apply a stochastic approach to systematically give gradient estimates for some important geometric quantities under the…

SGD and stochastic gradient descent converge at optimal rates for certain non-convex functions.

Study on geometric flows and rigidity of solitons.

The paper provides a geometric framework for understanding non-equilibrium thermodynamics.

In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equation \[ (Δ-q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this …

Gradient clipping helps private SGD converge despite potential bias.

The paper provides gradient estimates for a parabolic equation under Finsler geometric flows.

The study examines gradient almost Yamabe solitons in warped product manifolds and their geometric properties.

New flows introduced for symplectic geometry.

The paper proves various inequalities on gradient shrinking Ricci solitons.

GeoPhy uses geometric gradients to efficiently infer phylogenetic trees from molecular data.

This paper explores geometric and analytic aspects of Lojasiewicz inequalities on vector bundles.

GGA improves untrustworthy prediction detection in neural networks without retraining.

Stochastic gradient descent (SGD) is a key ingredient in the training of deep neural networks and yet its geometrical significance appears elusive. We study a deterministic model in which the trajectories of our dynamical systems are described via geodesics of a family of metrics arising from the diffusion matrix. Thes…

Geometric tempering improves sampling from distributions, with exponential convergence rates.

The paper classifies special geometric shapes in 2D and 3D.

Superconvergence of differential structure on discretized surfaces is studied in this paper. The newly introduced geometric supercloseness provides us with a fundamental tool to prove the superconvergence of gradient recovery on deviated surfaces. An algorithmic framework for gradient recovery without exact geometric i…

Geometric Occam's Razor shapes deep learning solutions.

We discuss some geometric conditions under which a complete noncompact shrinking gradient Ricci soliton will split at infinity.

SGD converges with perturbed forward-backward passes, explained by geometric amplification.

Unified framework for strain-gradient plasticity from dislocations.

SGLD proves geometric ergodicity via reflection coupling for nonconvex log-concave distributions.

In this paper, we extend the geometric descent method recently proposed by Bubeck, Lee and Singh to tackle nonsmooth and strongly convex composite problems. We prove that our proposed algorithm, dubbed geometric proximal gradient method (GeoPG), converges with a linear rate $(1-1/\sqrtκ)$ and thus achieves the optimal …

We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear stability of such gradient Ricci solitons.

Study on Einstein solitons with bounds and asymptotic behavior.

Study on gradient h-almost Yamabe solitons with scalar curvature estimation.

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of …

The paper introduces a geometric flow for Lagrangian submanifolds that preserves Hamiltonian isotopy.

BBVI with STL converges geometrically under perfect specification, with quadratic variance bound.

New geometric analysis shows $L^2$ score error is flawed for diffusion models.

In the present paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds.

Introduces new info-geometric structure for dynamics on graphs and hypergraphs.

The paper studies geometric properties of hydrodynamical density manifolds.

This paper addresses anisotropy in Transformer models, providing geometric insights and empirical support.

Byrd-SAGA reduces variance to robustify SGD against Byzantine attacks.

A lower-bound estimate of injectivity radius for complete Riemannian manifolds is discussed in a pure geometric viewpoint and is applied to study tangent cones at infinity of certain gradient Ricci solitons. We also study the asymptotic volume ratio of gradient Ricci solitons.

Researchers study solitons on homogeneous manifolds, proving properties of specific types of solitons.

Gradient descent converges to perfect classification in neural nets for non-separable data.

Paper shows how to use geometric median for robust SGD in high dimensions.

We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein met…

RIG extends IG to Riemannian manifolds for explainable AI.

Study on slow convergence in geometric variational problems.

Found first example of homogeneous gradient solitons for G$_2$-Laplacian flow.