arXiv research

In this paper we describe the notion of a weak lipschitzianity of a mapping on a $C^{q}$ stratification. We also distinguish a class of regularity conditions that are in some sense invariant under definable, locally Lipschitz and weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B) condition and the …

Bi-Lipschitz proof for 2-varifolds near critical Allard condition.

Maps between certain Lipschitz manifolds are isometries if they preserve volume.

New method for differentially private optimization with general Lipschitz conditions.

New method improves optimization algorithms without Lipschitz smoothness.

Proves conditions for Fourier transforms in rank 1 symmetric spaces.

We present sufficient conditions for the cohomology of a closed aspherical manifold to be proper Lipschitz in sense of Connes-Gromov-Moscovici [CGM]. The conditions are stated in terms of the Stone-Čech compactification of the universal cover of a manifold. We show that these conditions are formally weaker than the suf…

New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.

The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothn…

Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in $\rn$. The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smo…

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

Investigates intrinsic Lipschitz sections in nonlinear quotient maps.

Study maps in semidirect products of groups, proving Lipschitz properties without intrinsic dilations.

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

In this paper we prove that every definable set has a definable triangulation which is locally Lipschitz and weakly bi-Lipschitz on the natural simplicial stratification of the simplicial complex. We also distinguish a class T of regularity conditions and give a universal construction of a definable triangulation with …

New MIP formulations for neural network Lipschitz constant estimation.

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

In this paper, two sufficient conditions are provided for given two K-equivalent map-germs to be bi-Lipschitz A-equivalent. These are Lipschitz analogues of the known results on C^r-A-equivalence $(0 \leq r \leq \infty)$ for given two K-equivalent map-germs. As a corollary of one of our results, a Lipschitz version of …

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…

This is the first paper of two ones. Here we prove that two compact Alexandrov surfaces of bounded integral curvature having no peak points are bi-Lipschitz equivalent if they are homeomorphic one to the other. Also conditions under that two ends having finite integral negative curvature are bi-Lipschitz equivalent are…

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 local Lipschitz constant of a neural network is a useful metric with applications in robustness, generalization, and fairness evaluation. We provide novel analytic results relating the local Lipschitz constant of nonsmooth vector-valued functions to a maximization over the norm of the generalized Jacobian. We prese…

New scalable Lipschitz bounds improve neural network robustness analysis.

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…

In this paper we consider Llarull's theorem in the foliation case and get a lower bound of the Lipschitz constant of the map $M\to S^n$ in the foliation case under the spin condition.

New shuffling methods improve convergence without Lipschitz smoothness.

We investigate the challenge of multi-output learning, where the goal is to learn a vector-valued function based on a supervised data set. This includes a range of important problems in Machine Learning including multi-target regression, multi-class classification and multi-label classification. We begin our analysis b…

The paper examines bi-Lipschitz triviality of function germs on singular varieties.

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 …

In this paper, we study the convergence of generative adversarial networks (GANs) from the perspective of the informativeness of the gradient of the optimal discriminative function. We show that GANs without restriction on the discriminative function space commonly suffer from the problem that the gradient produced by …

The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.

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

We consider the problem of finding sufficient conditions for a locally Lipschitz mapping between Finsler manifolds to be a global homeomorphism. For this purpose, we develop the notion of Clarke generalized differential in this context and, using this, we obtain a version of the Hadamard integral condition for invertib…

The study establishes equivalence of conditions on metric manifolds with finite volume.

We prove that any finite dimensional Alexandrov space with a lower curvature bound is locally Lipschitz contractible. As applications, we obtain a sufficient condition for solving the Plateau problem in an Alexandrov space considered by Mese and Zulkowski.

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

In the article the necessary and sufficient conditions for a representation of Lipschitz function of two variables as a difference of two convex functions are formulated. An algorithm of this representation is given. The outcome of this algorithm is a sequence of pairs of convex functions that converge uniformly to a p…

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much …

Paper defines topology automaton for Barański carpets and proves Hölder equivalence conditions.

Introduces intrinsically Lipschitz graphs in metric spaces.

In this paper, we investigate the underlying factor that leads to failure and success in the training of GANs. We study the property of the optimal discriminative function and show that in many GANs, the gradient from the optimal discriminative function is not reliable, which turns out to be the fundamental cause of fa…

New parameterization of neural networks with Lipschitz bounds for robustness.

The goal of the paper is to design sequential strategies which lead to efficient optimization of an unknown function under the only assumption that it has a finite Lipschitz constant. We first identify sufficient conditions for the consistency of generic sequential algorithms and formulate the expected minimax rate for…

We give sufficient conditions for a $C^1_c$-local diffeomorphism between Fréchet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fréchet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a functio…

Lipschitz RNNs improve stability and performance in various tasks.

ResNets can approximate input distances under certain conditions, but existing theory is flawed.