arXiv research

New method tightens Lipschitz bounds for CNNs efficiently.

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

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

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…

Bi-Lipschitz flows approximate a wide range of distributions.

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 …

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

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

Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.

CLIP controls neural network stability by bounding Lipschitz constants.

Smooth solutions found for a specific type of Yamabe problem.

The paper proves Lipschitz regularity of graph Laplacian eigenvectors on random data clouds.

Characterizes quasi-isometric embeddings in coarsely Lipschitz category.

Adaptive LR improves neural network Lipschitz regularity without slowing convergence.

The paper quantifies the regularity of attention operations.

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.

We investigate robustness of deep feed-forward neural networks when input data are subject to random uncertainties. More specifically, we consider regularization of the network by its Lipschitz constant and emphasize its role. We highlight the fact that this regularization is not only a way to control the magnitude of …

The paper uses neural networks to learn system dynamics from data with Lipschitz regularization.

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(Ω, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…

Bi-Lipschitz Autoencoder ensures robust manifold preservation.

In this work we study input gradient regularization of deep neural networks, and demonstrate that such regularization leads to generalization proofs and improved adversarial robustness. The proof of generalization does not overcome the curse of dimensionality, but it is independent of the number of layers in the networ…

Enhances SSL with mixup and Lipschitz regularization.

Proves regularity for quasilinear elliptic equations in metric spaces.

We provide a characterization of r-regular sets in terms of the Lipschitz regularity of normal vector fields to the boundary.

Introduces intrinsically Lipschitz graphs in metric spaces.

Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.

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

The paper proposes a method to train NNs with a small Lipschitz constant to improve robustness.

New Lipschitz de Rham theorem for $L_p$-cohomology.

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

We determine regularity results for energy minimizing maps from an $n$-dimensional Riemannian polyhedral complex $X$ into a CAT(1) space. Provided that the metric on $X$ is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the doma…

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…

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 …

We augment adversarial training (AT) with worst case adversarial training (WCAT) which improves adversarial robustness by 11% over the current state-of-the-art result in the $\ell_2$ norm on CIFAR-10. We obtain verifiable average case and worst case robustness guarantees, based on the expected and maximum values of the…

Generative algorithms learn high-dimensional data efficiently and generate new samples.

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

A map between manifolds is an isometry if it's Lipschitz and scalar curvature bounded.

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for o…

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…

Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial res…

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

New framework enhances neural network robustness against adversarial attacks.

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…

Improved robustness of machine learning models with controlled Lipschitz constants.

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.

Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.

Since their invention, generative adversarial networks (GANs) have become a popular approach for learning to model a distribution of real (unlabeled) data. Convergence problems during training are overcome by Wasserstein GANs which minimize the distance between the model and the empirical distribution in terms of a dif…

Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.