The H1(ds)-gradient flow shrinks circles with radius r0 to a point.
problem The triviality of the L2(ds) metric topology on immersed planar curves. method Gradient flow of the length functional with respect to the H1(ds)-metric. result Circles shrink to a point under the H1(ds)-gradient flow. New method for constrained sampling using gradient flows.
problem Sampling from constrained domains.
method Introducing a boundary condition for gradient flow to confine particles within the domain.
result Provable continuous-time convergence in total variation for constrained sampling.
New particle-based VI algorithm expands function class and improves scalability.
problem Limited function class in particle-based VI algorithms restricts flexibility and scalability.
method Introduces a functional regularization term to expand the function class and proposes PFG algorithm.
result Proposed PFG algorithm has larger function class, improved scalability, better adaptation to ill-conditioned distributions, and provable convergence.
Gradient flow studies Spin(7)-structures on compact 8-manifolds.
problem Formulating and studying the gradient flow of Spin(7)-structures.
method Negative gradient flow of an energy functional of Spin(7)-structures.
result Short-time existence and uniqueness of solutions to the flow.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
Proves Thom's conjecture for parabolic flows on Hilbert spaces.
problem Gradient flows on infinite-dimensional spaces and geometric flows with symmetry.
method Analytic functions, Hilbert spaces, Yang-Mills Flow, Ricci flow, critical points, Lojasiewicz inequality.
result Gradient conjecture holds for parabolic flows on Hilbert spaces, including flows with gauge symmetry.
Paper proves Harnack inequality for f-mean curvature flow.
problem Proving Harnack inequality for f-mean curvature flow. method Gradient flow of the weighed area functional with measure density function e−f. result Proves Li-Yau-Hamilton type Harnack estimate.
Constructs retractions of CAT(1) spaces to convex subsets.
problem Geometric description of an analytic tool.
method Gradient flow of time-dependent locally Lipschitz semiconcave functions.
result Existence of gradient flows proved for independent interest.
The paper defines and analyzes higher-order Yang-Mills-Higgs functionals and their gradient flows.
problem Analyzing the behavior of higher-order Yang-Mills-Higgs functionals and their gradient flows.
method Gauge fixing technique, L2-bound of the Higgs field, local L2-derivative estimates, energy estimates, blow-up analysis. result Solutions to the gradient flow do not hit finite time singularities under certain conditions.
Self-test loss functions improve data-driven modeling of weak-form operators and gradient flows.
problem Challenges in selecting test functions for data-driven modeling involving weak-form operators and gradient flows.
method Introducing self-test loss functions that depend on unknown parameters and are quadratic.
result Self-test loss functions conserve energy for gradient flows and coincide with log-likelihood ratios for stochastic differential equations.
SIFG uses noisy particles to efficiently sample from complex distributions.
problem Efficient sampling from complex distributions using particle-based methods.
method SIFG introduces a semi-implicit functional gradient flow with Gaussian noise to improve sampling efficiency and accuracy.
result SIFG achieves strong theoretical convergence guarantees and efficient sampling.
Gradient flow on diffeomorphisms for image registration, with well-posedness proven.
problem Image registration with metric tensor deformation penalization.
method Gradient flow on Sobolev diffeomorphisms for a specific energy functional.
result Well-posedness of the gradient flow established.
Gradient flows on graphons converge to curves on graphon space.
problem Optimizing functions on large, exchangeable graphs.
method Euclidean gradient flow on edge weights converges to a curve on graphon space.
result Gradient flows on graphons can be described as curves of maximal slope on graphon space.
The paper studies a flow to prescribe curvature on CR manifolds.
problem Prescribing the $ar{Q}'$-curvature on three dimensional Pseudo-Einstein CR manifolds.
method Gradient flow generated by a related functional.
result Convergence of the flow to a limit function under suitable assumptions.
Gradient flows of neural networks converge to optimal values or diverge, with thresholds and asymptotic behaviors.
problem Understanding the convergence and divergence of gradient flows in neural networks.
method Analysis of gradient flows on loss landscapes of neural networks using o-minimal structures.
result Gradient flows either converge to optimal values or diverge to infinity, with thresholds and asymptotic behaviors.
Proves Arnold-Thom conjecture for surfaces' arrival times.
problem Existence of limit tangents for gradient flow lines of surfaces.
method Gradient flow lines of mean curvature flows with neck or cylindrical singularities.
result Proves Arnold's conjecture for all mean convex mean curvature flows of surfaces.
This work develops a particle system to approximate Fisher-Rao gradient flows in mean-field optimization.
problem Optimizing probability measures in neural network contexts.
method Constructing an interacting particle system approximating Fisher-Rao gradient flows.
result Propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization.
Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.
problem Whether mean curvature flow is a gradient flow on nondegenerate metric spaces of simple closed plane curves.
method Examined two nondegenerate metric spaces: uniformness-preserving and curvature-weighted structures.
result Mean curvature flow is not a gradient flow on either metric space.
We study the L2 gradient flow of the Yang--Mills functional on the space of connection 1-forms on a principal G-bundle over the sphere S2 from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space ΩG of based loops in the compact Lie group G. An iso…
We consider the flows generated by generic gradients of Morse maps of a closed connected manifold M to a circle. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called…
Gradient flow solves multi-index regression for high-dimensional Gaussian data.
problem Learning multi-index functions from high-dimensional Gaussian data.
method Two-timescale algorithm with non-parametric link function learning.
result Global convergence of Grassmannian population gradient flow dynamics.
Gradient flow method solves for optimal transport starting distributions.
problem Finding the optimal starting distribution for a martingale in optimal transport.
method Following the gradient flow of the Bass functional's L2-lift.
result Gradient flow converges to a minimizer of the Bass functional.
This paper explores gradient flows for sampling distributions without normalization constants.
problem Sampling from distributions with unknown normalization constants.
method Gradient flows in the space of probability measures, focusing on Kullback-Leibler divergence, Fisher-Rao metric, and affine invariance.
result Gradient flows derived from Kullback-Leibler divergence do not depend on the normalization constant.
A new ParVI framework improves particle-based variational inference methods.
problem Non-trivial kernel design in particle-based variational inference methods.
method Proposes a generalized Wasserstein gradient descent (GWG) framework with broader regularizers.
result Demonstrates strong convergence guarantees and effectiveness on simulated and real data.
Identifies a gradient flow to solve kernel learning problems with noise reduction.
problem Kernel learning problem with Gaussian noise.
method Riemannian gradient flow with continuous Lyapunov functionals.
result Flow reduces noise and finds stationary points.
For a harmonic function u on Euclidean space, this note shows that its gradient is essentially determined by the geometry of its level hypersurfaces. Specifically, the factor by which |grad(u)| changes along a gradient flow is completely determined by the mean curvature of the level hypersurfaces intersecting the flow.
New spinorial functional connects Perelman's W- and F-functionals.
problem Unifying Perelman's functionals for spin manifolds.
method Introduced a new energy functional on spin manifolds, computed its first variation, and established a gradient flow.
result Critical points of the functional are twisted Ricci solitons and eigen-spinsors.
On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, φ) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor φ. We investigate the basic properties of this functional and study its negative gradient f…
The paper derives inequalities and formulas for generalized Ricci flow.
problem Understanding and characterizing generalized Ricci flow.
method Using Bochner formula and adapted Malliavin gradient, the paper derives inequalities and characterizes generalized Ricci flow.
result Characterizations of generalized Ricci flow via inequalities for the associated Malliavin gradient.
Gradient flow expands curves to round shapes.
problem Expanding curves to round shapes.
method Steepest descent L2-gradient flow of entropy.
result Flow converges to a round expanding circle for various initial curves.
Study homotopy equivalence of spaces of gradient-like flows and Morse functions on surfaces.
problem Understanding the topology of spaces of smooth functions and flows on surfaces.
method Proves homotopy equivalence of spaces of gradient-like flows and Morse functions on surfaces, with detailed decomposition into orbits.
result Spaces of gradient-like flows and Morse functions on surfaces are homotopy equivalent to manifolds.
Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting
problem Fluctuations of boosting processes around their deterministic limit
method Stochastic perturbation analysis of ODEs in Banach spaces
result Rescaled deviations converge to a Gaussian process
The paper studies critical points and flows of a G2-Hilbert functional on manifolds with circle actions.
problem Critical points and flows of the G2-Hilbert functional on manifolds with S1-actions. method Analysis of S1-invariant G2-structures, reduction to a 6-dimensional quotient, and derivation of a negative L2-gradient flow. result The unnormalized flow admits only trivial stationary configurations: flat connection, scalar-flat base metric, and constant fiber length.
Optimizes convex functions in finite vs infinite dimensions, revealing slow convergence rates.
problem Analyzing gradient flows in finite and infinite-dimensional Hilbert spaces.
method Proves convergence rates and optimality conditions for gradient flows and related methods.
result Gradient flow convergence rates in finite dimensions are slower than in infinite dimensions, with optimal rates achievable in Hilbert spaces.
Gradient flow of curve length on Sobolev metrics preserves convexity.
problem Optimal low-regularity gradient flow of curve length.
method Explicit gradient formula, Picard-Lindelöf theorem, time-reparametrisation.
result Exponential decay of length and preservation of convexity.
This paper bridges variational inference and Wasserstein gradient flows.
problem Combining variational inference and Wasserstein gradient flows for more efficient approximations.
method Recasting Bures-Wasserstein gradient flow as a Euclidean gradient flow and using path-derivative gradient estimator.
result A new gradient estimator for f-divergences that can be implemented using machine learning libraries. We present a novel approximate inference method for diffusion processes, based on the Wasserstein gradient flow formulation of the diffusion. In this formulation, the time-dependent density of the diffusion is derived as the limit of implicit Euler steps that follow the gradients of a particular free energy functional.…
Examples of Morse functions with integrable gradient flows on some classical Riemannian manifolds are considered. In particular, we show that a generic height function on the symmetric embeddings of classical Lie groups and certain symmetric spaces is a perfect Morse function, i.e. has as many critical points as the ho…
Gradient descent on MMD GAN parameter space converges globally to target distribution.
problem Convergence of gradient descent in Maximum Mean Discrepancy (MMD) GANs.
method Proposes a parametric kernelized gradient flow that mimics the min-max game in gradient regularized MMD GAN.
result Gradient descent on the generator's parameter space in gradient regularized MMD GAN is globally convergent to the target distribution under certain conditions.
Forward-Euler fails for simulating Wasserstein gradient flows with KL divergence.
problem Simulating Wasserstein gradient flows with forward-Euler discretization fails for KL divergence.
method Forward-Euler discretization for Wasserstein gradient flows with KL divergence.
result Forward-Euler discretization can be incorrect for Wasserstein gradient flows with KL divergence.
In his 2011 work, Maas has shown that the law of any time-reversible continuous-time Markov chain with finite state space evolves like a gradient flow of the relative entropy with respect to its stationary distribution. In this work we show the converse to the above by showing that if the relative law of a Markov chain…
In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riem…
Morse theory connects low energy submanifolds in 3-sphere.
problem Understanding low energy submanifolds in the 3-sphere.
method Morse-theoretic techniques and negative gradient flow.
result Constructs connections between low energy critical submanifolds.
We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinkin…
The Sinkhorn flow converges to a Wasserstein mirror gradient flow from the Sinkhorn algorithm.
problem Optimizing joint distributions using the Sinkhorn algorithm.
method Wasserstein mirror gradient flow derived from the Sinkhorn algorithm.
result The Sinkhorn flow converges to a Wasserstein mirror gradient flow.
The paper describes flows of MMD functionals with distance kernel and quantile functions.
problem Wasserstein gradient flows of MMD functionals with negative distance kernel.
method Characterization via Cauchy problem on L2(0,1), solution via subdifferential construction. result Flow invariance and smoothing properties on subsets of C(0,1), absolute continuity of initial measures. We analyze a gradient flow of closed planar curves minimizing the anisoperimetric ratio. For such a flow the normal velocity is a function of the anisotropic curvature and it also depends on the total interfacial energy and enclosed area of the curve. In contrast to the gradient flow for the isoperimetric ratio, we sho…
Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.
problem Analysis of geometric properties on asymmetric metric measure spaces.
method Introduction of upper gradients, q-Laplacian, and q-heat flow in asymmetric settings. result Extension of concepts from symmetric to asymmetric metric measure spaces.