arXiv research

Uniformly extend maps on Hadamard manifolds with curvature constraints.

Uniform Lipschitz continuity of isoperimetric profiles in evolving surfaces.

Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.

Proposes a weaker version of Strong Cosmic Censorship with curvature bounds.

Guarantees uniform convergence for square-root Lipschitz losses.

This paper refines homotopy theory for cubical sets and uniform spaces.

New method for differentially private optimization with general Lipschitz conditions.

The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.

The study examines metrics on Riemannian spaces with bounded properties and finds conditions for Lipschitz and uniform bounds.

The paper proves Rademacher's theorem for Heisenberg groups.

Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

Counterexamples show failure of uniform laws of large numbers for subdifferentials.

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Ru…

Improved DP SO with large Lipschitz parameters, handling outliers and heavy-tailed data.

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale …

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary codimension for L less than or equal to 1/4.

Method approximates Lipschitz domains with smoother shapes.

The study shows that certain graphs are regular at boundary points.

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

We provide an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear e…

Unified framework for uniform signal recovery in nonlinear GCS with 1-bit/quantized measurements.

Extends Lipschitz functions while preserving local constants.

Paper introduces risk assessment for contextual bandits without experiments.

We study controlled systems which are uniformly observable and differentially observable with an order larger than the system state dimension. We establish that they may be transformed into a (partial) triangular canonical form but with possibly non locally Lipschitz functions. We characterize the points where this Lip…

Uniform covers with a finite-dimensional nerve are rare (i.e., do not form a cofinal family) in many separable metric spaces of interest. To get hold on uniform homotopy properties of these spaces, a reasonably behaved notion of an infinite-dimensional metric polyhedron is needed; a specific list of desired properties …

Let $\H^n$ be the Heisenberg group of topological dimension $2n+1$. We prove that if $n$ is odd, the pair of metric spaces $(\H^n, \H^n)$ does not have the Lipschitz extension property.

Existence and rigidity results for lifts in Carnot groups.

New bound relaxes uniform gradient norm assumptions for PAC-Bayesian bounds.

Study on residual Monge-Ampère mass of complex functions with directional Lipschitz continuity.

Unified high-probability regret bounds for online convex optimisation with randomised gradient estimators.

In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…

This paper examines weight initialization for 1-Lipschitz networks to improve robustness against adversarial attacks.

Let $Σ$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:Σ\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean standard metric. We prove a general curvature obstruction to the existence of merely differen…

We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm that induces d. There is an Out(F_n)-invariant potential Ψon Outer space such that when the Lipschitz norm is corrected by the derivative of Ψ, the resulting norm is quasisymmetric. As an application, we give new …

The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.

Bi-Lipschitz mappings can embed certain algebraic sets into high-dimensional spaces.

A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over…

Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.

We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stron…

Paper relaxes the Lipschitz constraint in WGANs to improve performance.

For all $k,n\ge 1$, we construct a biLipschitz embedding of $\mathbb{S}^n$ into the jet space Carnot group $J^k(\mathbb{R}^n)$ that does not admit a Lipschitz extension to $\mathbb{B}^{n+1}$. Let $f:\mathbb{B}^n\to \mathbb{R}$ be a smooth, positive function with $k^{th}$-order derivatives that are approximately linear …

New rigidity theorem for product of lattices.

Let us consider a Riemannian manifold $M$ (either separable or non-separable). We prove that, for every $ε>0$, every Lipschitz function $f:M\rightarrow\mathbb R$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le \Lip(f)+ε$. As a consequence, every Riemannian manifold is uniformly …

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we provide a technique which transfers results on uniform approximation on bounded …

RVFL networks can efficiently approximate Lipschitz functions in L∞ norm.

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…

Extends growth properties of hyperbolic groups to their extensions.