arXiv research

Abstract: Lipschitz homeomorphisms are deformed using Perelman's methods.

Lipschitz RNNs improve stability and performance in various tasks.

New Transformers maintain Lipschitz continuity for robustness.

Neural nets learn robust geometric data representations.

CLIP controls neural network stability by bounding Lipschitz constants.

Kim-Milman flow map stable under regular target measures

New method tightens Lipschitz bounds for CNNs efficiently.

Maps preserving mass and injective on boundary are isometries.

Study examines stability of image-reconstruction algorithms using variational regularization.

Study shows stability of travel time data reconstruction from closed subsets.

The paper proves topological stability between RCD spaces and Riemannian manifolds.

We introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spac…

Study quantifies convergence of Alexandrov spaces without collapsing.

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

The paper stabilizes invertible neural networks by using Gaussian mixture models.

Study shows how to control jump-diffusion processes with stable feedback controls in reinforcement learning.

An $e^ε$-Lipschitz and co-Lipschitz map, as a metric analogue of an $ε$-Riemannian submersion, naturally arises from a sequence of Alexandrov spaces with curvature uniformly bounded below that converges to a space of only weak singularities. In this paper we prove its homotopy lifting property and its homotopy stabilit…

The paper trains neural networks with robustness guarantees using semidefinite constraints.

Optimally estimates stability in Lorentzian isoperimetric inequalities.

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…

We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the general…

Generative adversarial networks (GANs) are one of the most popular approaches when it comes to training generative models, among which variants of Wasserstein GANs are considered superior to the standard GAN formulation in terms of learning stability and sample quality. However, Wasserstein GANs require the critic to b…

New scalable Lipschitz bounds improve neural network robustness analysis.

Push-forward models struggle to fit multimodal distributions due to high Lipschitz constants.

Given a free factor A of the rank n free group F_n, we characterize when the subgroup of Out(F_n) that stabilizes the conjugacy class of A is distorted in Out(F_n). We also prove that the image of the natural embedding of Aut(F_{n-1}) in Aut(F_n) is nondistorted, that the stabilizer in Out(F_n) of the conjugacy class o…

Global stability bounds for matrix frames in phase retrieval problems.

Let $Ω\subset \mathbb{R}^3$ be a Lipschitz domain, and consider a harmonic map $v: Ω\rightarrow \mathbb{S}^2$ with boundary data $v|\partialΩ= \varphi$ which minimises the Dirichlet energy. For $p\geq 2$, we show that any energy minimiser $u$ whose boundary map $ψ$ has a small $W^{1,p}$-distance to $\varphi$ is close t…

Study limits of curved spaces with boundaries.

The Lookahead optimizer improves SGD's performance and generalization without restrictive assumptions.

Framework selects optimal historical data windows for non-stationary learning.

Study Ricci-Deturck flow from rough metrics, proving short-time existence.

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

The goal of this paper is to study the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. We prove that if two metrics on a $n$-manifold of bounded sectional curvature are $L_0$-bi-Lipchitz equivalent and sufficient collapsed (depending on $L_0$ and $n$), then up to a diffeo…

The paper studies stability of mean-field variational inference for log-concave distributions.

The paper proves LOO CV is reliable under estimator stability.

Recover simple irreversible Finsler geometry from travel time data

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its $C^2$-distance fro…

Tight estimation of the Lipschitz constant for deep neural networks (DNNs) is useful in many applications ranging from robustness certification of classifiers to stability analysis of closed-loop systems with reinforcement learning controllers. Existing methods in the literature for estimating the Lipschitz constant su…

Study on Gaussian interpolation flows for generative modeling.

Study tackles inverse problems on low-dimensional manifolds, proving stability and proposing a reconstruction algorithm.

New bounds show faster convergence for learning algorithms.

Stability of Dehn functions proven for ultralimits of Sobolev maps.

Copula-based normalizing flows improve flexibility and stability for heavy-tailed data.

Normalization layers control deep neural network capacity, improving stability and generalization.

CGNNs use wavelets for continuous function generation in infinite-dimensional spaces.

Develops a framework for distilling flow models from few steps.

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

ARBITER learns SPX-VIX term structures without arbitrage constraints.