arXiv research

The paper studies properties of intrinsically Lipschitz constants in metric spaces.

New metric measure space theory for Lipschitz constants.

Paper presents an efficient algorithm for estimating Lipschitz functions from noisy data.

Investigates intrinsic Lipschitz sections in nonlinear quotient maps.

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

We focus our attention on the notion of intrinsic Lipschitz graphs, inside a special class of metric spaces i.e. the Carnot groups. More precisely, we provide a characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2 in terms of their intrinsic distributional gradients.

In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respe…

Introduces intrinsically Lipschitz graphs in metric spaces.

The paper proves Rademacher's theorem for Heisenberg groups.

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

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

Maps in Carnot groups are equivalent to solutions of a PDE system.

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

Criterion for surfaces in Heisenberg group to be graphs using flat cones.

We prove that Lipschitz intrinsic graphs in the Heisenberg groups $H^n$, with $n>1$, which are vanishing viscosity solutions of the minimal surface equation are smooth.

The paper studies maps in the Heisenberg group and their images, called Rickman rugs.

Enhances neural networks' robustness against adversarial samples without sacrificing clean sample generalization.

We prove that, in the first Heisenberg group $\mathbb{H}$, an entire locally Lipschitz intrinsic graph admitting vanishing first variation of its sub-Riemannian area and non-negative second variation must be an intrinsic plane, i.e., a coset of a two dimensional subgroup of $\mathbb{H}$. Moreover two examples are given…

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

Paper analyzes a new Hopf-Lax semigroup in metric spaces.

Deep networks can approximate functions with fewer learnable parameters than previously thought.

We compute the local Lipschitz constant of ReLU networks precisely.

New approach to certifiably robust neural networks using Boolean function perspective.

In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φ_y+ [φ^{2}/2]_t=w, where w is a bounded function depending o…

New MIP formulations for neural network Lipschitz constant estimation.

Paper investigates Lipschitz constants of self-attention modules in neural networks.

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

Extends Lipschitz functions while preserving local constants.

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

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

Several recent papers have discussed utilizing Lipschitz constants to limit the susceptibility of neural networks to adversarial examples. We analyze recently proposed methods for computing the Lipschitz constant. We show that the Lipschitz constant may indeed enable adversarially robust neural networks. However, the m…

The paper studies harmonic graphs in the Heisenberg group and their properties.

This paper analyzes the Lipschitz constants of deep neural networks with random weights.

This paper studies bounds for the Lipschitz constant of random neural networks.

This work focuses on important step in quantitative topology: given homotopic mappings from $S^m$ to $S^n$ of Lipschitz constant $L$, build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant). The present paper resolves this problem for the first case where Hopf invariant p…

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

The study proves surfaces in a specific Heisenberg group must be simple planes.

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

LiPopt uses polynomial optimization to estimate neural network Lipschitz constants efficiently.

New method for differentially private optimization with general Lipschitz conditions.

Study shows stable graphs in Heisenberg group are essentially planes.

The paper proposes a least squares method for binary compressive sampling with low intrinsic dimension signals.

CLIP controls neural network stability by bounding Lipschitz constants.

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

We determine the asymptotic behavior of the optimal Lipschitz constant for the systole map from Teichmuller space to the curve complex.

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 calculates upper bounds on ReLU network Lipschitz constants.

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 $…