Optimizes shapes of curves using Möbius energy gradients.
problem Finding optimal shapes of curves within isotopy classes.
method Gradient-based optimization with Sobolev inner products.
result Significantly more efficient and robust optimization methods.
Sobolev training helps neural nets fit function values and derivatives.
problem Training neural nets to match function values and derivatives accurately.
method Using Sobolev loss with gradient flow for overparameterized networks.
result Gradient flow from random initialization can fit any function and its derivatives.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
Gradient descent achieves optimal learning for elliptic PDEs via Sobolev norms.
problem Learning elliptic PDEs from noisy data.
method Gradient descent on Sobolev norm objective functions.
result Gradient descent achieves statistical optimality for elliptic PDEs.
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.
The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
problem Gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
method Assumes a Sobolev inequality and integral Ricci bounds, proving local gradient estimates and Liouville type results.
result Proves local gradient estimates and Liouville type results on manifolds with lower bounds of Ricci curvature.
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 estimates for special harmonic functions on manifolds.
problem Estimating gradients of (p,V)-harmonic functions on Riemannian manifolds. method Using Moser iteration method, volume comparison theorem, and Sobolev embedding theorem.
result Explicit global gradient estimates for positive entire (p,V)-harmonic functions. Paper introduces SGA for barycenter optimization in optimal transport.
problem Optimizing Wasserstein barycenter for discrete distributions.
method Sobolev gradient ascent algorithm tailored to Wasserstein geometry.
result SGA achieves convergence rate similar to subgradient descent.
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.
Proves inequality linking function deviation to gradient norm on compact manifolds.
problem Analyzing coupled elliptic systems on compact manifolds.
method Develops a new Poincaré-Sobolev inequality with a density-free reference average.
result Poincaré constant depends on the density's gradient norm.
The paper analyzes how adding gradient data affects overparameterized models' performance.
problem The impact of Sobolev training on high-dimensional predictive models.
method Combining replica method from statistical physics and operator-valued free probability theory.
result Sobolev training does not universally improve performance for target functions described by single-index models.
New framework improves EM algorithm convergence under log-Sobolev inequality.
problem Improving convergence of the EM algorithm.
method Extending gradient flow techniques to EM algorithm, using free energy representation.
result Exponential convergence of EM algorithm under log-Sobolev inequality.
The paper provides approximation guarantees for neural networks trained with gradient flow.
problem Approximating neural networks trained with gradient flow in continuous L2(Sd−1)-norm. method NTK argument for non-convex second but last layer, under-parametrized regime.
result Gradient flow convergence guarantees for neural networks under Sobolev smoothness assumptions.
New algorithm samples superlinearly growing log-gradient distributions.
problem Sampling from distributions with superlinearly growing log-gradient.
method Proposes a novel taming Langevin-based scheme called sTULA.
result Derives non-asymptotic convergence bounds in KL, TV, and W2 distances.
The study establishes inequalities for functions on manifolds using Green function estimates.
problem Developing inequalities for functions on manifolds.
method Used integral representations and uniform estimates for Green functions.
result Proved Lp Sobolev-type and Poincaré-type inequalities for functions on real and complex manifolds. Study on extremizers for Sobolev inequality on curved manifolds.
problem Existence of extremizers for the sharp p-Sobolev inequality on Riemannian manifolds with nonnegative curvature. method Nonsmooth concentration compactness methods and Mosco-convergence results for Cheeger energy.
result Almost extremal functions are close to radial Euclidean bubbles and almost zero globally under nonnegative curvature.
New method estimates densities using Sobolev regularization, outperforming existing algorithms.
problem Non-parametric density estimation with clear inductive bias.
method Regularizes Sobolev norm of density, approximates kernel via sampling, uses natural gradients for optimization.
result Method ranks second best on ADBench anomaly detection benchmark.
We study a simplification of GAN training: the problem of transporting particles from a source to a target distribution. Starting from the Sobolev GAN critic, part of the gradient regularized GAN family, we show a strong relation with Optimal Transport (OT). Specifically with the less popular dynamic formulation of OT …
The paper provides gradient estimates for solutions on manifolds with integral Ricci bounds.
problem Global regularity estimates for solutions of Δu=f on Riemannian manifolds. method Proves Lp-gradient estimates under integral Ricci bounds and constructs a counterexample. result Optimal constant lower bounds on Ricci curvature are shown in the pointwise sense.
A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension m in Rn with the space of flat m-cochains, that is, the dual space of flat chains of dimension m in Rn. The main purpose of the present paper is to generalize Wolfe's theorem to the se…
Study nonexistence and gradient estimates for solutions on manifolds with bounded Ricci curvature.
problem Nonexistence and gradient estimates for solutions of a specific quasi-linear equation on manifolds.
method Utilizes Sobolev inequalities and geometric properties to establish results.
result Extends and improves previous results on nonexistence and gradient estimates.
Proves error bounds for PGD, extending log-Sobolev and Talagrand inequalities.
problem Maximum likelihood estimation of large latent variable models.
method Extending log-Sobolev and Talagrand inequalities to models with strongly concave log-likelihoods.
result Non-asymptotic error bounds for PGD in models satisfying LSI and PŁI.
Develops theory linking Schrödinger equations to manifold ends, proving finiteness.
problem Understanding the number of ends in Riemannian manifolds.
method Variant of Li-Tam theory, polynomial growth analysis, Sobolev inequality.
result Finiteness of manifold ends under scaling invariant Sobolev inequality.
Paper introduces a new regularization method for kernel gradient descent learning.
problem Preventing overfitting in kernel gradient descent learning.
method Random smoothing regularization as novel convolution-based smoothing kernels.
result Optimal convergence rates achieved in various function spaces.
We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…
We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the L2 Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.
The study establishes inequalities on path space for sub-Riemannian manifolds.
problem Understanding functional inequalities on path space for sub-Riemannian manifolds.
method Derivative and integration by parts formulae on path space with respect to a natural gradient operator, showing bounds of horizontal Ricci curvature.
result Established functional inequalities on path space analogous to Riemannian geometry.
We show that the concept of H2-gradient flow for the Willmore energy and other functionals that depend at most quadratically on the second fundamental form is well-defined in the space of immersions of Sobolev class W2,p from a compact, n-dimensional manifold into Euclidean space, provided that p≥2 and…
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
The paper discusses fractional Sobolev immersions of flat domains into 3D space.
problem Developing C1 regularity and isometric immersions of flat domains with fractional Sobolev regularity. method Analysis of weak Codazzi-Mainardi equations, study of $W^{2, rac2s}$ planar deformations, and properties of the distributional Jacobian determinant.
result Generalization of isometric immersions with local fractional Sobolev regularity.
Paper proposes new Langevin samplers for sampling from log-concave distributions with superlinear gradient growth.
problem Sampling from log-concave distributions with superlinear gradient growth.
method Proposes two novel discretizations of kinetic Langevin SDEs, showing contractivity and log-Sobolev inequality.
result Establishes non-asymptotic bounds in 2-Wasserstein distance between sampled distributions and target measures.
New bound relaxes uniform gradient norm assumptions for PAC-Bayesian bounds.
problem Generalization bounds with strict assumptions like uniformly bounded loss.
method Relax uniform bounds assumptions to on-average bounded loss and gradient norm.
result Proposes a new generalization bound with a surrogate of model complexity.
In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves…
NTK neural networks are robust to adversarial attacks in nonparametric regression.
problem Adversarial robustness of neural networks in nonparametric regression.
method Gradient flow with early stopping for NTK neural networks, proving robustness in Sobolev spaces.
result NTK neural networks achieve optimal adversarial robustness rates in Sobolev spaces.
Scheme minimizes p-elastic energy of curves over time.
problem Minimizing p-elastic energy of curves over time. method Minimizing movement scheme with approximate normal graphs.
result Short-time existence and lower bound on solution's lifetime.
Gradient flow of elastic energy converges to elastica.
problem Optimizing closed curves to minimize elastic energy.
method Proving the existence of a unique global solution and convergence via Łojasiewicz--Simon gradient inequality.
result Convergence to elastica established for the H2(ds)-gradient flow of modified elastic energy. Variational methods for revealing visual concepts learned by convolutional neural networks have gained significant attention during the last years. Being based on noisy gradients obtained via back-propagation such methods require the application of regularization strategies. We present a mathematical framework unifying…
Paper relaxes the Lipschitz constraint in WGANs to improve performance.
problem WGANs do not always outperform other GAN variants due to imperfect implementation of the Lipschitz condition.
method Proposes a new dual form of Wasserstein distance (Sobolev duality) that relaxes the Lipschitz constraint but maintains gradient property.
result SWGAN, based on Sobolev duality, outperforms existing methods in experiments.
A function that optimally aligns a timelike vector field with its gradients
problem Finding a time function that aligns a timelike vector field with its gradients
method Introducing a functional that penalizes null gradients and minimizes misalignment
result Proving the existence of a unique alignment time function under suitable conditions
The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchhei…
The paper establishes Sobolev inequalities between Riemannian metrics and their distance functions.
problem Establishing a theory of Sobolev inequalities for Riemannian metrics and distance functions.
method Analyzing the sub-critical case $p < rac{m}{2}$, proving a Sobolev inequality linking $L^{ rac{p}{2}}$ bounds on metrics to Lq bounds on distance functions. result A Sobolev inequality exists between Riemannian metrics and their distance functions, leading to a convergence theorem.
We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient …
Sharp upper diameter limit found for Ricci solitons.
problem Bounding the diameter of compact shrinking Ricci solitons.
method Used a sharp logarithmic Sobolev inequality and Vitali-type covering argument.
result Sharp upper diameter bound established in terms of scalar curvature and entropy.
Paper improves convergence rate of Langevin Dynamics algorithms.
problem Sampling problems and non-convex optimization in machine learning.
method Stochastic Variance Reduced Gradient Langevin Dynamics and Stochastic Recursive Gradient Langevin Dynamics with improved convergence rates.
result Proves convergence to objective distribution under weaker conditions.
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.
Lower bounds for eigenvalues on Bakry-Emery manifolds proven.
problem Eigenvalue estimates on Bakry-Emery manifolds.
method Generalised maximum principle and heat kernel estimates.
result Lower bounds for all eigenvalues proven.
Proposes a new generalization bound for Bayesian deep nets without strict assumptions.
problem Lack of generalization bounds for Bayesian deep nets without strict assumptions.
method Exploits contractivity of Log-Sobolev inequalities to add a loss-gradient norm term to the generalization bound.
result Introduces a new generalization bound for Bayesian deep nets that avoids strict assumptions.