53 problems

Thom's gradient conjecture. If $\gamma(t)\to x_0\in\mathcal{U}$, then $\gamma$ has a tangent at $x_0$; equivalently, the limit of secants

Let $f$ be a real analytic function on an open set $U\subset\mathbb{R}^n$, and let $x(t)$ be a gradient flow line of $f$ with a limit point $x_0\in U$. Arnold–Thom conjecture. If…

Let $\Omega$ be a convex domain, and consider a gradient flow in $\Omega$ whose arising path has length $L$. The gradient-flow length conjecture. There is a universal constant $c$…

Let $X$ be a compact $3$-manifold, and let $gc(X)$ denote its gradient complexity, as introduced in the paper. A nonsingular gradient flow on $X$ is a gradient flow without singula…

Let ${\mathbf P}$ be the space of probability densities under consideration, let $k$ be a viscosity function, and let ${\mathcal D}^k$ and $\widetilde{\mathcal D}^k$ be the distanc…

Carlier and Poon's minimum-principle conjecture. The minimum principle holds for the TV-JKO scheme in every dimension $d$.

Avoidance conjecture. If a compact connected component $C$ of $\operatorname{Crit}(f)$ consists of weakly strict saddle points, then $W^s_-(C)$ has measure zero. Moreover, if $M$ i…

The paper considers generalized gradient flows and convex optimization on entanglement polytopes in Hadamard manifolds. In particular, the preceding discussion uses unitary matrice…

Consider the one-dimensional Cauchy problem for equation … should hold for every … This conjecture concerns the range of parameters for which positive weak solutions exist; the pap…

The energy adaptive method is developed for the Ginzburg–Landau energy. Gradient-flow conjecture. Its use on other gradient flow problems could alleviate issues for other complex c…

Deformation conjecture. Every smooth, non-self-intersecting $U\in{\mathcal A}$ can be smoothly deformed into $U_R$ along a path contained in ${\mathcal A}$.

Let $X \in \mathbb{R}^{n \times T}$ satisfy $C_X \succ 0$, and let $W(t)$ follow the continuous-time gradient-flow feedforward synaptic dynamics. Let $\mathcal{S}$ denote the stati…

Let $X \in \mathbb{R}^{n \times T}$ satisfy $C_X \succ 0$, and let $W(t)$ solve the continuous-time gradient-flow feedforward synaptic dynamics … Write $\mathcal{R}_{m,n}$ for the…

Let $\Delta$ be the $\ell^2$-simplex equipped with the Fisher–Rao metric, and let the gradient flow lines of $F_{(c_n)}$ be given by the integral curves in Proposition. In finite-d…

Time-uniform stability conjecture. The fact that stability does not degrade over time is connected to the gradient-flow nature of the dynamics and the convergence of the flow to a…

Let $\mathcal{P}_{2,c}(\mathbb{R})$ be the space of centered probability measures on $\mathbb{R}$ with finite second moment, and for $\mu_0,\mu_1\in\mathcal{P}_{2,c}(\mathbb{R})$ d…

Let $gamma(t)$ be a gradient flow line for an analytic function, and suppose that it has a limit point. Arnold--Thom gradient conjecture. The limit … exists. This is a stronger, fi…

Concentration conjecture. For high dimensions $p$ and design matrices $\mathbf{X}$ that have limited long-range dependence across variables, the estimate

Let the minimizing movements scheme for the relaxed macroscopic IPM problem be defined by time steps of size $h>0$, with iterates obtained by minimizing the sum of the Wasserstein-…

Let $U$ be the functional on $mathcal{P}2(mathbb{R}^d)$ defined by $U(rho)=frac{1}{2}W2^2(rho,e^{-g})$, and for absolutely continuous probability measures $xi$ and $rho$ let … whe…

Consider the assumptions imposed in the paper to prove the tight asymptotic convergence rate for the Fisher–Rao gradient flow of the Kullback–Leibler divergence and the analogous r…

Let $(\rho_t^{\mathrm{WFR}})_{t\geq 0}$ denote the Wasserstein–Fisher–Rao gradient flow of the Kullback–Leibler divergence, and let the relevant expansion refer to the explicit asy…

The posterior distribution $\rho_{\rm post}(\theta)$ is strongly log-concave, and the convergence-rate constants for the Gaussian approximate gradient flow and Gaussian approximate…

Let $\rho_0\in\mathcal{P}_V$, let $S_r$ be the stated class of flows, and for $u\in S_r$ define the pushforward measure $\rho_t=u(t,\cdot)_{\#}\rho_0$. For $u$-regularized Stein F…

Let $\rho_t$ denote the solution of the Langevin Fokker–Planck equation with initial density $\rho_0$, and let the target density $\pi$ satisfy the log-Sobolev inequality (LSI). As…