Tootfinder

Hessian matrix estimates of heat-type equations via Bismut-Stroock Hessian formula
Li-Juan Cheng, Rui-Yu Yang
https://arxiv.org/abs/2506.11805 https://

Hessian matrix estimates of heat-type equations via Bismut-Stroock Hessian formula
In this paper, we establish a new global Hessian matrix estimate for heat-type equations on Riemannian manifolds using a Bismut-type Hessian formula. Our results feature fully explicit coefficients as well as delay / growth rate functions. These estimates yield two key applications: a novel backward weak Harnack inequality and a precise pointwise Hessian estimate for eigenfunctions.

Heavy-ball dynamics with Hessian-driven damping for non-convex optimization under the {\L}ojasiewicz condition
Vassilis Apidopoulos, Vasiliki Mavrogeorgou, Theodoros G. Tsironis
https://arxiv.org/abs/2506.11705

Heavy-ball dynamics with Hessian-driven damping for non-convex optimization under the Łojasiewicz condition
In this paper, we examine the convergence properties of heavy-ball dynamics with Hessian-driven damping in smooth non-convex optimization problems satisfying a Łojasiewicz condition. In this general setting, we provide a series of tight, worst-case optimal convergence rate guarantees as a function of the dynamics' friction coefficients and the Łojasiewicz exponent of the problem's objective function. Importantly, the linear rates that we obtain improve on previous available rates and they sug…

Exponential Vorticity Hessian Growth in Capillary Liquid Drop in Two Dimensions
Zhongtian Hu
https://arxiv.org/abs/2506.11414 https://

Exponential Vorticity Hessian Growth in Capillary Liquid Drop in Two Dimensions
In this work, we concern ourselves with the evolution of a droplet of an ideal fluid in two dimensions, which has nontrivial bulk vorticity and is only subject to the effects of surface tension. We construct initial data with initial domain being a disk and initial velocity being arbitrarily small, such that the vorticity Hessian grows exponentially infinitely in time.

A continuous-time model to interpolate between speed and function value restart in accelerated first order methods
Juan Jos\'e Maul\'en, Huiyuan Guo, Juan Peypouquet
https://arxiv.org/abs/2506.11267

A continuous-time model to interpolate between speed and function value restart in accelerated first order methods
We introduce a new restarting scheme for a continuous inertial dynamics with Hessian driven-damping, and establish a linear convergence rate for the function values along the restarted trajectories. The proposed routine is a generalization of existing speed restart schemes, and interpolates between speed and function value restarts, considerably delaying the restarting time, while preserving the function value decrease. Numerical experiments show an improvement in the convergence rates for both…

Numerical stability of force-gradient integrators and their Hessian-free variants in lattice QCD simulations
Kevin Sch\"afers, Jacob Finkenrath, Michael G\"unther, Francesco Knechtli
https://arxiv.org/abs/2506.08813

Numerical stability of force-gradient integrators and their Hessian-free variants in lattice QCD simulations
We investigate the numerical stability of force-gradient integrators and their Hessian-free variants within the molecular dynamics step of the Hamiltonian Monte Carlo algorithm in lattice QCD simulations. A linear stability analysis of (Hessian-free) force-gradient integrators is conducted by investigating the harmonic oscillator as a test equation. By performing detailed stability investigations for the entire family of self-adjoint integrators with up to eleven exponentials per time step, we …

This https://arxiv.org/abs/2407.12518 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Inertial Methods with Viscous and Hessian driven Damping for Non-Convex Optimization
In this paper, we aim to study non-convex minimization problems via second-order (in-time) dynamics, including a non-vanishing viscous damping and a geometric Hessian-driven damping. Second-order systems that only rely on a viscous damping may suffer from oscillation problems towards the minima, while the inclusion of a Hessian-driven damping term is known to reduce this effect without explicit construction of the Hessian in practice. There are essentially two ways to introduce the Hessian-driv…

This https://arxiv.org/abs/2309.09432 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Entire solutions of two-convex Lagrangian mean curvature flows
Given an entire $C^2$ function $u$ on $\mathbb{R}^n$, we consider the graph of $D u$ as a Lagrangian submanifold of $\mathbb{R}^{2n}$, and deform it by the mean curvature flow in $\mathbb{R}^{2n}$. This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove long-time existence and convergence results under a 2-positivity assumption of $(I+(D^2 u)^2)^{-1}D^2 u$. Such results were previously known only under the stronger assumption of positivity of $D^2 u…

GaussianVAE: Adaptive Learning Dynamics of 3D Gaussians for High-Fidelity Super-Resolution
Shuja Khalid, Mohamed Ibrahim, Yang Liu
https://arxiv.org/abs/2506.07897

GaussianVAE: Adaptive Learning Dynamics of 3D Gaussians for High-Fidelity Super-Resolution
We present a novel approach for enhancing the resolution and geometric fidelity of 3D Gaussian Splatting (3DGS) beyond native training resolution. Current 3DGS methods are fundamentally limited by their input resolution, producing reconstructions that cannot extrapolate finer details than are present in the training views. Our work breaks this limitation through a lightweight generative model that predicts and refines additional 3D Gaussians where needed most. The key innovation is our Hessian-…

On the role of the Parity Violating Hojman--Holst term in Gravity Theories
Damianos Iosifidis
https://arxiv.org/abs/2506.04738 https://

On the role of the Parity Violating Hojman--Holst term in Gravity Theories
We study Parity Violating Gravity Theories whose gravitational Lagrangian is a generic function of the scalar curvature and the parity odd curvature pseudoscalar, commonly known as the Holst (or Hojmann) term. Generalizing some previous results in the literature, we explicitly show that if the Hessian of this function is non-degenerate, the initial non-Riemannian Theory is on-shell equivalent to a metric Scalar-Tensor Theory. The generic form of the kinetic coupling function and the scalar pote…

This https://arxiv.org/abs/2503.14353 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Unified Analysis of Decentralized Gradient Descent: a Contraction Mapping Framework
The decentralized gradient descent (DGD) algorithm, and its sibling, diffusion, are workhorses in decentralized machine learning, distributed inference and estimation, and multi-agent coordination. We propose a novel, principled framework for the analysis of DGD and diffusion for strongly convex, smooth objectives, and arbitrary undirected topologies, using contraction mappings coupled with a result called the mean Hessian theorem (MHT). The use of these tools yields tight convergence bounds, b…

General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications
Jiabin Yin, Xingjian Zhou
https://arxiv.org/abs/2506.01434

General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications
In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities. In particular, when $k=1$, we give an alternative proof for Brandolini-Nitsch-Salani's results \cite{BNS}. Meanwhile…

This https://arxiv.org/abs/2502.03701 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

First-ish Order Methods: Hessian-aware Scalings of Gradient Descent
Gradient descent is the primary workhorse for optimizing large-scale problems in machine learning. However, its performance is highly sensitive to the choice of the learning rate. A key limitation of gradient descent is its lack of natural scaling, which often necessitates expensive line searches or heuristic tuning to determine an appropriate step size. In this paper, we address this limitation by incorporating Hessian information to scale the gradient direction. By accounting for the curvatur…

The twisted constant in Calabi-Yau type equation
Genglong Lin
https://arxiv.org/abs/2506.09639 https://arxiv.org/pdf/2506.09639

The twisted constant in Calabi-Yau type equation
In this paper we establish a necessary and sufficient condition for solving a general class of fully nonlinear elliptic equations on compact almost Hermitian manifolds, extending a recent work of Guo-Song. As applications, we determine the twisted constants in Calabi-Yau type equations, including the classical one, Hessian type one and form type one with gradient terms introduced by Popovici and Tosatti-Weinkove. In particular, it addresses a question raised in a work of Chu-Tosatti-Weinkove \c…

The Minkowski problem for the $k$-torsional rigidity
Xia Zhao, Peibiao Zhao
https://arxiv.org/abs/2505.24494 https://arxiv.org/pdf/25…

The Minkowski problem for the $k$-torsional rigidity
P. Salani [Adv. Math., 229 (2012)] introduced the $k$-torsional rigidity associated with a $k$-Hessian equation and obtained the Brunn-Minkowski inequalities $w.r.t.$ the torsional rigidity in $\mathbb{R}^3$. Following this work, we first construct, in the present paper, a Hadamard variational formula for the $k$-torsional rigidity with $1\leq k\leq n-1$, then we can deduce a $k$-torsional measure from the Hadamard variational formula. Based on the $k$-torsional measure, we propose the Minkowsk…

Progressively Projected Newton's Method
Jos\'e Antonio Fern\'andez-Fern\'andez, Fabian L\"oschner, Jan Bender
https://arxiv.org/abs/2505.21013

Progressively Projected Newton's Method
Newton's Method is widely used to find the solution of complex non-linear simulation problems in Computer Graphics. To guarantee a descent direction, it is common practice to clamp the negative eigenvalues of each element Hessian prior to assembly - a strategy known as Projected Newton (PN) - but this perturbation often hinders convergence. In this work, we observe that projecting only a small subset of element Hessians is sufficient to secure a descent direction. Building on this insight, we…

This https://arxiv.org/abs/2306.14853 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Near-Optimal Nonconvex-Strongly-Convex Bilevel Optimization with Fully First-Order Oracles
In this work, we consider bilevel optimization when the lower-level problem is strongly convex. Recent works show that with a Hessian-vector product (HVP) oracle, one can provably find an $ε$-stationary point within ${\mathcal{O}}(ε^{-2})$ oracle calls. However, the HVP oracle may be inaccessible or expensive in practice. Kwon et al. (ICML 2023) addressed this issue by proposing a first-order method that can achieve the same goal at a slower rate of $\tilde{\mathcal{O}}(ε^{-3})$. In this pap…