Questions tagged [gradient-flows]
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42 questions
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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 ...
8
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0
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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 ...
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2
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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 ...
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6
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1
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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 ...
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3
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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 ...
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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 ...
5
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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 ...
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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. $$\dot{x_t} = - \...
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Gradient flows: convex potential vs. contractive flow?
Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$).
Consider the autonomous gradient-flow
$$
\dot ...
- 5,380
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Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$
In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
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3
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1
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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)&...
- 311
3
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1
answer
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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 ...
3
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1
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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
$$
\dot{x}(t) = -\nabla g(x(t)), \;...
- 6,853
3
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1
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Gradient flows: evolution of geodesics
I’m trying to understand if, when I move the marginals of a Wasserstein geodesic along a contractive flow, the geodesic between the new probability measures is “near” to the geodesic
connecting the ...
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3
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1
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Equivalent definition of the Kantorovich-Fisher-Rao distance
I am reading this paper
"A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows"
(https://arxiv.org/abs/1602.04457)
and in the proof of Proposition 2.2, basically, if the measure ...
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1
answer
483
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Flow induced by differentiable velocity field is differentiable
Let $E$ be a $\mathbb R$-Banach space, $\tau>0$ and $v:[0,\tau]\times E\to E$ such that$^1$ $$x\mapsto t\mapsto v(t,x)\tag1$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$. This is enough to ensure ...
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0
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Harmonic heat flow, formal and rigorous
Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of
$$
\partial_tu-\Delta ...
- 1,219
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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(\...
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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}...
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Gradient flows and particle representations
I was looking into gradient flows and their particle representations, mostly in the context of probability.
A simple example of this is the continuity equation. Consider evolving a sample $x \sim \...
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1
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308
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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 ...
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2
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1
answer
142
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Existence of first variation
I am trying to compute the first variation of the functional
$$\mathcal F(\rho) = \int_{\Omega} R(x;\rho) d\rho(x)$$
where $R$ is some function of $x$ that also depends on $\rho$. Here $\rho$ is a ...
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1
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677
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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 ...
- 3,098
2
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1
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291
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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 ...
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2
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0
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views
Is it likely that gradient flow trajectories of a $G$-invariant function pass through degenerate points?
This question may be posed somewhat vaguely, but I'm interested to actually get an idea of what to expect, so I try to not target it at a specific result.
Assume that $G$ is a compact Lie group, ...
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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.
...
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2
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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))...
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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 ...
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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 "...
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vote
1
answer
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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., $f(gm) = f(m)$ ...
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1
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Is it true that $\xi \in \partial G (v)$ implies $\frac{\xi}{F'(\phi (v))} \in \partial \phi (v)$?
I am reading the introduction of Chapter 10 in the book Gradient Flows by Ambrosio and his coauthors.
As we have seen in Section 1.4, in the classical theory of subdifferential calculus for proper, ...
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1
answer
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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 ...
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vote
1
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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,+\...
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1
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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 $\mathbb{R}^d$...
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0
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154
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 $\nabla g:\...
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Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures
TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures.
Let $(X, d)$ ...
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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
$$\begin{cases}\mathbf{u}'(t)\in-\partial ...
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0
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259
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Expressing the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary
I want to express the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary. For this I use the Einstein-Hilbert action
$$S(g_{\mu \nu})=\frac{1}{16\pi}\...
- 1,303
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254
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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?
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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 $\...
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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 $\mathbb{T}^2$.
...
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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 ...
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