# Questions tagged [gradient-flows]

The gradient-flows tag has no usage guidance.

37
questions

-1
votes

0
answers

34
views

### Does the non-existence of second order critical point implies the convergence of gradient flow?

Given a dynamical system
$\dot \theta_i=-\sum_{j=1}^N\sin(\theta_i-\theta_j)$
where $\vec\theta\in\mathbb{R}^N$. Note that the system is a gradient system $\dot{\vec\theta}=-\sum_{i<j}\cos(\theta_i-...

2
votes

1
answer

248
views

### When does uniqueness of a stable equilibrium imply it is globally stable?

Given a gradient dynamical system
$$\dot x=-\nabla f(x),$$
my question is:
(1) If there exists only one equilibrium $x^*$ which is stable (if necessary, this can be changed to stable asymptotically ...

1
vote

1
answer

129
views

### Gradient descent under the presence of symmetries

Let $M$ be a Riemannian manifold (I'm happy to assume it is Euclidean space) with a function $f: M \to \mathbb R$ and a group of isometries $G$ acting on $M$ and preserving $f$, i.e., $f(gm) = f(m)$ ...

2
votes

1
answer

193
views

### What are the best definitions for smoothness of a 2D curve (real-valued function)?

Sounds like a trivial question, but could not find any answer other than the fact that there are many ways to define it. My problem is this: I look at different elevation maps,
some with sharp ...

0
votes

0
answers

35
views

### On deriving gradient flow equations from a given energy functional

I am reading quite a lot of references and am new to the connection between optimal transport and stochastic differential equations. In Topics in Optimal Transportation Villani 8.3.1, there are quite ...

5
votes

1
answer

185
views

### How can we prove that a stochastic process converges to a deterministic value?

As an illustrative example, consider a modified O-U process $dX_t = -X_tdt + \exp(-t)dW_t$. It is not too hard to understand that after a while the behaviour is dominated by the deterministic ...

1
vote

0
answers

147
views

### Relation between two gradient dynamics

If $f:\mathbb{R}^n\rightarrow\mathbb{R}_+$ is a nonnegative real analytic function and $g:\mathbb{R}^n\rightarrow\mathbb{R}$ is a strongly convex smooth function with a surjective gradient $\nabla g:\...

1
vote

0
answers

69
views

### Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures

TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures.
Let $(X, d)$ ...

2
votes

1
answer

146
views

### Gradient descent relaxation dynamics of a Euler-Lagrange equation

I want to minimize the functional
$$
F=\int{L(u)}dx,
$$
where $L= u_x^2-u^2$ is the Lagrangian function of the functional. Even if its Euler-Lagrange equation is easily found and solved, I want to try ...

3
votes

1
answer

153
views

### Does gravity constant affect boundedness of solution?

Consider a second order gradient-like system with linear damping
$$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$
Suppose $f\in C^2(\mathbb{R}^n)$ and $\inf_{x\in\mathbb{R}^n}f(x)&...

3
votes

1
answer

110
views

### Equivalent definition of the Kantorovich-Fisher-Rao distance

I am reading this paper
"A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows"
(https://arxiv.org/abs/1602.04457)
and in the proof of Proposition 2.2, basically, if the measure ...

3
votes

0
answers

70
views

### Regularity of center manifold

Consider a $C^r$ vector field $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with $r \geq 1$. Let $\bar x$ be a critical point of $f$, that is, $f(\bar x) = 0$.
Suppose that the spectrum of $\mathrm{D}f(\...

0
votes

0
answers

191
views

### Geodesics and gradient flow

Is there a construction in Riemannian geometry which relates the gradient flow of a function on a manifold with a certain metric with geodesics on another related manifold with its own metric?

6
votes

2
answers

441
views

### The negative gradient flow of a Morse-Bott function on a compact manifold converges to a critical point?

Let $(M, g)$ be a compact Riemannian manifold and $f: M \rightarrow \mathbb{R}$ be a Morse-Bott function, i.e. the set a critical points of $f$, $Crit(f)$, has connected components which are smooth ...

0
votes

0
answers

164
views

### Convergence of ODE solutions almost everywhere to a stable equilibrium point

Theorem: Suppose ${\bf g} :\mathbb{R}^n \mapsto \mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A} \subset \mathbb{R}^n$ such that $\bf g$ is uniformly Lipschitz on $\...

2
votes

0
answers

223
views

### Do Chern-Simons terms qualitatively alter the behavior of the Yang-Mills gradient flow?

I'm reading about the Yang-Mills heat flow, and I'm curious how adding a Chern-Simons term alters its solutions. This is probably elementary or folklore, but I don't know well enough to say.
...

0
votes

1
answer

314
views

### Rewriting PDE as "push-forward"

Suppose that we have the following PDE
$$\partial_t \mu_t = \nabla\cdot \left(\nabla \mu_t - (b*\mu_t)\mu_t\right), \tag{1}$$
with $\mu_0$ being a (smooth) probability measure/density on $\mathbb{R}^d$...

3
votes

1
answer

369
views

### Geometric flow by the level sets of a harmonic function

Let $u$ be an harmonic function in a cylindrical domain $B_2^{n-1}\times(-1,1)\subset\mathbb{R}^n$, and suppose its level sets $\Gamma_t=\{u=t\}$ are graphs of functions on $B_2^{n-1}$.
Consider a ...

2
votes

0
answers

56
views

### Chain recurrent points of a gradient-like system

Let $X$ be a compact metric space and $f:X\to X
$ homeomorphism. Let $V:X\to \mathbb{R}$ be a Lyapunov function for $(X,f)$ (continuous function such that $(\forall x\notin Fix(f))\ \ V(f(x))<V(x))...

1
vote

1
answer

157
views

### Gradient-like dynamical systems

I've tried asking this question on Mathematics site, but I only got an upvote and no answer. I've searched online, tried to find something about this topic, but I haven't found much (and the things I ...

6
votes

1
answer

189
views

### Intersection of self-shrinkers

I have a problem regarding a statement in the paper Smooth compactness of self-shrinkers by Colding and Minicozzi.
In the article, they define a surface $\Sigma$ in $\mathbb R^3$ to be a self-shrinker ...

5
votes

0
answers

118
views

### What kinds of gradient-flows on $\mathbb R^d$ preserve the log-concavity of the distribution $\mu_0$ of starting point $x_0$

Let $\mu_0$ be a log-concave distribution on $\mathbb R^d$ and let $f:\mathbb R^d \to \mathbb R$ be $C^2$. Let $x_0$ be sampled uniformly at random from a log-concave distribution $\mu_0$, meaning ...

2
votes

0
answers

47
views

### A question about strong slopes (nonsmooth analysis)

Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time ...

0
votes

0
answers

142
views

### Metric obstructions for area-preserving diffeomorphisms with constant singular values

Let $\mathbb{T}^2$ be the topological $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$. Let $g$ be an arbitrary smooth Riemannian metric on $\mathbb{T}^2$.
...

1
vote

1
answer

180
views

### Metric / strong slope restriction of function on unit ball in $\mathbb R^m$

Diclaimer. I'm not sure this is the right venue for this question, but I'll give it a try
Definition [Strong / metric slope]. Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+\...

5
votes

3
answers

582
views

### What quantities are conserved under a general gradient-flow $\dot X(t) = -\nabla L(X(t))$?

Let $L:\mathbb R^N \to \mathbb R$ be a continuously differential function with gradient $x \mapsto \nabla L(x)$ and consider induced gradient-flow
$$
\dot X(t) = -\nabla L(X(t)).
$$
Question. Is ...

3
votes

1
answer

199
views

### Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field

Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow
$$
\dot{x}(t) = -\nabla g(x(t)), \;...

2
votes

0
answers

82
views

### Concentration inequalities for gradient flows induced by random fields

Let $G=(G(x))_{x \in \mathbb R^m}$ be a conservative random field with values in $\mathbb R^m$, for large positive integer $m$. That is, there exists a scalar random field $g=(g(x))_{x \in \mathbb R^m}...

0
votes

0
answers

97
views

### Relation between test and train error with gradient descent iterates

My question is about establishing an inequality between population error and expected training error (i.e, expected training error < population error) for a model trained with gradient descent on a ...

2
votes

1
answer

418
views

### Flow induced by differentiable velocity field is differentiable

Let $E$ be a $\mathbb R$-Banach space, $\tau>0$ and $v:[0,\tau]\times E\to E$ such that$^1$ $$x\mapsto t\mapsto v(t,x)\tag1$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$. This is enough to ensure ...

2
votes

0
answers

65
views

### Center-stable manifold theorem on manifold with boundary

I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary.
Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...

3
votes

2
answers

584
views

### Gradient flows: convex potential vs. contractive flow?

Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$).
Consider the autonomous gradient-flow
$$
\dot ...

8
votes

0
answers

275
views

### What is nice in gradient flows?

First of all, I am sorry for such naivety. I have not all the intuition in hard analysis as I wish.
I am studying Perelman's work and his big first contribution is to prove that the Ricci flow is in ...

4
votes

0
answers

239
views

### Basin of attraction of gradient flow

Suppose we have a compact Riemannian manifold $(M,g)$, and a Morse function $f : M \rightarrow \mathbb{R}$. Suppose we consider the gradient flow generated by this function, i.e. $$\dot{x_t} = - \...

1
vote

0
answers

149
views

### Well-posedness of gradient flows

For a convex lower-semicontinuous functional on a Hilbert space $I\colon H\rightarrow\mathbb{R}$, it is shown in Evans' PDE that the Hilbert-space-valued ODE
$$\begin{cases}\mathbf{u}'(t)\in-\partial ...

1
vote

0
answers

247
views

### Expressing the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary

I want to express the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary. For this I use the Einstein-Hilbert action
$$S(g_{\mu \nu})=\frac{1}{16\pi}\...

11
votes

2
answers

2k
views

### Textbooks or notes on gradient flows in metric spaces

What is a good introduction in gradient flows in metric spaces?
I know the book Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe ...