Related papers: On gradient flows initialized near maxima

We consider maximum solution $g(t)$, $t\in [0, +\infty)$, to the normalized Ricci flow. Among other things, we prove that, if $(M, \omega) $ is a smooth compact symplectic 4-manifold such that $b_2^+(M)>1$ and let $g(t),t\in[0,\infty)$, be…

We study the behavior of averages for functions defined on finite graphs $G$, in terms of the Hardy-Littlewood maximal operator $M_G$. We explore the relationship between the geometry of a graph and its maximal operator and prove that $M_G$…

We generalize the gradient flow equation for field theories with nonlinearly realized symmetry. Applying the formalism to super Yang-Mills theory, we construct a supersymmetric extension of the gradient flow equation. It can be shown that…

A flow defined by a nonsingular smooth vector field $X$ on a closed manifold $M$ is said to be parameter rigid if given any real valued smooth function $f$ on $M$, there are a smooth funcion $g$ and a constant $c$ such that $f=X(g)+c$…

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal…

This paper investigates the application of mini-batch gradient descent to semiflows (gradient flows). Given a loss function (potential), we introduce a continuous version of mini-batch gradient descent by randomly selecting sub-loss…

One of the most popular approaches for solving total variation-regularized optimization problems in the space of measures are Particle Gradient Flows (PGFs). These restrict the problem to linear combinations of Dirac deltas and then perform…

A special case of the main result states that a complete $1$-connected Riemannian manifold $(M^n,g)$ is isometric to one of the models $\mathbb R^n$, $S^n(c)$, $\mathbb H^n(-c)$ of constant curvature if and only if every $p\in M^n$ is a…

This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in…

We disclose an interesting connection between the gradient flow of a $\mathcal{C}^2$-smooth function $\psi$ and evanescent orbits of the second order gradient system defined by the square-norm of $\nabla\psi$, under adequate convexity…

Given a `cost' functional $F$ on paths $\gamma$ in a domain $D\subset\mathbb{R}^d$, in the form $F(\gamma) = \int_0^1 f(\gamma(t),\dot\gamma(t))dt$, it is of interest to approximate its minimum cost and geodesic paths. Let $X_1,\ldots, X_n$…

Given a random sample from a density function supported on a manifold $M$, a new method for the estimating highest density regions of the underlying population is introduced. The new proposal is based on the empirical version of the opening…

This paper is a survey of the generalized Hamiltonian gradient flow (GHGF) framework for Hamilton-Jacobi equations, with an emphasis on the propagation of singularities and its connections to weak KAM theory, optimal transport and mean…

We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of…

We consider the $L^2$ gradient flow for the Willmore functional in Riemannian manifolds of bounded geometry. In the euclidean case E.\;Kuwert and R.\;Sch\"atzle [\textsl{Gradient flow for the Willmore functional,} Comm. Anal. Geom., 10:…

We solve the conjecture posed by Fill, Scheinerman and Singer-Cohen and show the equivalence of the sharp threshold functions of the random intersection graph G(n,m,p) with $m >= n^3$ and a graph in which each edge appears independently.…

Existing analyses of optimization in deep learning are either continuous, focusing on (variants of) gradient flow, or discrete, directly treating (variants of) gradient descent. Gradient flow is amenable to theoretical analysis, but is…

We introduce a new parabolic flow deforming any Riemannian metric on a spin manifold by following a constrained gradient flow of the total scalar curvature. This flow is built out of the well-known Dirac-Einstein functional. We prove local…

This paper addresses the gradient flow -- the continuous-time representation of the gradient method -- with the smooth approximation of a non-differentiable objective function and presents convergence analysis framework. Similar to the…

We investigate length decreasing maps $f:M\to N$ between Riemannian manifolds $M$, $N$ of dimensions $m\ge 2$ and $n$, respectively. Assuming that $M$ is compact and $N$ is complete such that…