Related papers: Gradient flow of the norm squared of a moment map

We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we…

Let $(M,g)$ be a closed Riemannian manifold, and let $F:M \to \mathbb{R}$ be a smooth function on $M$. We show the following holds generically for the function $F$: for each maximum $p$ of $F$, there exist two minima, denoted by $m_+(p)$…

The Teichm\"uller harmonic map flow, introduced in [9], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy…

A fully coupled system of two second-order parabolic degenerate equations arising as a thin film approximation to the Muskat problem is interpreted as a gradient flow for the 2-Wasserstein distance in the space of probability measures with…

We prove that the standard gradient flow in parameter space that underlies many training algorithms in deep learning can be continuously deformed into an adapted gradient flow which yields (constrained) Euclidean gradient flow in output…

We develop a geometric flow framework to investigate two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by height-stretching…

This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparameterized two-layer neural networks. In particular, we consider the minimax optimization problem stemming from estimating linear…

We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.

This paper extends previous work from arxiv:1702.05223, which shows that the main theorem of Morse theory holds for a large class of functions on singular spaces, where the function and the underlying singular space are required to satisfy…

In this note, we prove that the abstract gradient flow introduced by Baird-Fardoun-Regbaoui \cite{BFR}is well-posed on a closed Riemann surface with conical singularity. Long time existence and convergence of the flow are proved under…

The Teichm\"uller harmonic map flow deforms both a map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the map as quickly as possible…

We study the convergence properties of gradient descent for training deep linear neural networks, i.e., deep matrix factorizations, by extending a previous analysis for the related gradient flow. We show that under suitable conditions on…

How can we understand gradient-based training over non-convex landscapes? The edge of stability phenomenon, introduced in Cohen et al. (2021), indicates that the answer is not so simple: namely, gradient descent (GD) with large step sizes…

We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of {\it capacitary measures} with respect to the…

A recent line of work has shown remarkable behaviors of the generalization error curves in simple learning models. Even the least-squares regression has shown atypical features such as the model-wise double descent, and further works have…

We study the Wasserstein gradient flow of the Sinkhorn divergence when both the source and the target are Gaussian distributions. We prove the existence of a flow that stays in the class of Gaussian distributions, and is unique in the…

Uncertainty propagation and filtering can be interpreted as gradient flows with respect to suitable metrics in the infinite dimensional manifold of probability density functions. Such a viewpoint has been put forth in recent literature, and…

Sampling a target probability distribution with an unknown normalization constant is a fundamental challenge in computational science and engineering. Recent work shows that algorithms derived by considering gradient flows in the space of…

We derive the system of differential equations for the gradient flow characterizing the training process of linear in-context learning in full generality. Next, we explore the geometric structure of the gradient flows in two instances,…

In this paper, we introduce a class of new logarithmic curvature flow. The flows are designed to embrace the monotonicity of the related functional, and the convergence of this flow would tackle the solvability of the weighted…