arXiv research

Constructs a map with prescribed local Lipschitz constants on a subset of a manifold.

The paper proves Lipschitz continuity of cut times in spacetimes.

Proves existence of curved surfaces in hyperbolic space.

We show that the isoperimetric profile $h_{g(t)}(ξ)$ of a compact Riemannian manifold $(M,g)$ is jointly continuous when metrics $g(t)$ vary continuously. We also show that, when $M$ is a compact surface and $g(t)$ evolves under normalized Ricci flow, $h^2_{g(t)}(ξ)$ is uniform Lipschitz continuous and hence $h_{g(t)}(…

New methods solve MI problems with locally Lipschitz operators, improving solution efficiency.

Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity …

We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.

A generalization of the Flow-box Theorem is given. The assumption of continuous differentiability of the vector field is relaxed to a local Lipschitz condition. The theorem holds in any Banach space.

The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations…

New method improves optimization algorithms without Lipschitz smoothness.

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${…

Lorentzian distances to Cauchy surfaces fail to be locally equi-Lipschitz.

We prove local Holder continuity of quasi-n-harmonic mappings from Euclidean domains into metric spaces with non-positive curvature in the sense of Alexandrov. We also obtain global Holder continuity of such mappings from bounded Lipschitz domains.

MLDL preserves manifold geometry in vector transformations.

New methods for convex optimization with locally Lipschitz gradient, achieving faster convergence.

Extends involutivity to non-Lipschitz subbundles and proves the Frobenius Theorem.

New framework tightens certified robustness gaps in machine learning models.

Proposes DMOC for more nuanced neural network robustness.

A scattering transform defines a signal representation which is invariant to translations and Lipschitz continuous relatively to deformations. It is implemented with a non-linear convolution network that iterates over wavelet and modulus operators. Lipschitz continuity locally linearizes deformations. Complex classes o…

This paper bounds the Lipschitz constants of neural networks and their gradients.

Study 2D spaces with curvature, focusing on structure and approximations.

New Transformers maintain Lipschitz continuity for robustness.

Probabilistic learning is increasingly being tackled as an optimization problem, with gradient-based approaches as predominant methods. When modelling multivariate likelihoods, a usual but undesirable outcome is that the learned model fits only a subset of the observed variables, overlooking the rest. In this work, we …

The paper examines convergence of distances in Lipschitz structures on manifolds.

GroupSort neural networks can approximate Lipschitz continuous functions.

The curse of dimensionality affects neural network optimization, especially with smooth functions.

The paper studies Lipschitz bounds for integral kernels under differentiability assumptions.

We provide a new local class-purity theorem for Lipschitz continuous DNN classifiers. In addition, we discuss how to achieve classification margin for training samples. Finally, we describe how to compute margin p-values for test samples.

Investigates Lipschitz continuity in neural networks across various settings.

OTAD uses optimal transport to create robust models against adversarial attacks.

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…

This research explores principles of Lipschitz continuity in neural networks for robustness and generalization.

Study Lipschitz regularity for manifold-constrained ROF model on curved surfaces.

We examine the impact of learning Lipschitz continuous models in the context of model-based reinforcement learning. We provide a novel bound on multi-step prediction error of Lipschitz models where we quantify the error using the Wasserstein metric. We go on to prove an error bound for the value-function estimate arisi…

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $α:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(α(x))_k=|<x,f_k>|^2$, where $…

Let $(X,d,μ)$ be a complete metric measure space, with $μ$ a locally doubling measure, that supports a local weak $L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on $(X,d,μ)$. Gradient estimates for Cheeger-harmonic func…

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value …

Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.

Continuity of roots of hyperbolic polynomials with smooth coefficients.

Optimistic method adapted for faster convex-concave min-max problems.

The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

We provide a detailed proof of Hawking's singularity theorem in the regularity class $C^{1,1}$, i.e., for spacetime metrics possessing locally Lipschitz continuous first derivatives. The proof uses recent results in $C^{1,1}$-causality theory and is based on regularisation techniques adapted to the causal structure.

Extends Lipschitz functions while preserving local constants.

Study geometric flows with varying parameters and prove continuous dependence.

We model how Lipschitz continuity changes during neural network training.

Lipschitz continuity recently becomes popular in generative adversarial networks (GANs). It was observed that the Lipschitz regularized discriminator leads to improved training stability and sample quality. The mainstream implementations of Lipschitz continuity include gradient penalty and spectral normalization. In th…