arXiv research

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

We compute the local Lipschitz constant of ReLU networks precisely.

Extends Lipschitz functions while preserving local constants.

The paper calculates upper bounds on ReLU network Lipschitz constants.

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

The paper calculates bounds on the local Lipschitz constants of neural network layers.

HALO uses local Lipschitz constants to optimize functions efficiently.

Efficient local Lipschitz bounds improve neural network robustness.

Characterizes hypergenerated stratified groups with flat boundaries.

The paper extends a measure preserving property to bi-Lipschitz maps between Moran sets.

We study distributed optimization algorithms for minimizing the average of \emph{heterogeneous} functions distributed across several machines with a focus on communication efficiency. In such settings, naively using the classical stochastic gradient descent (SGD) or its variants (e.g., SVRG) with a uniform sampling of …

The Jacobian matrix (or the gradient for single-output networks) is directly related to many important properties of neural networks, such as the function landscape, stationary points, (local) Lipschitz constants and robustness to adversarial attacks. In this paper, we propose a recursive algorithm, RecurJac, to comput…

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

The paper proves Lipschitz continuity of cut times in spacetimes.

Techniques known as Nonlinear Set Membership prediction, Kinky Inference or Lipschitz Interpolation are fast and numerically robust approaches to nonparametric machine learning that have been proposed to be utilised in the context of system identification and learning-based control. They utilise presupposed Lipschitz p…

New MIP formulations for neural network Lipschitz constant estimation.

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

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

New metric measure space theory for Lipschitz constants.

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

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

The paper sharpens a theorem about surfaces with zero Gaussian curvature.

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…

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…

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

We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Further…

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

New method for differentially private optimization with general Lipschitz conditions.

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.

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 $…

Scalable method bounds Lipschitz constant of generative models.

Let $ρ: G \rightarrow \operatorname{O}(V)$ be a real finite dimensional orthogonal representation of a compact Lie group, let $σ= (σ_1,\ldots,σ_n) : V \to \mathbb R^n$, where $σ_1,\ldots,σ_n$ form a minimal system of homogeneous generators of the $G$-invariant polynomials on $V$, and set $d = \max_i \operatorname{deg} …

Efficiently learns Single-Index Models with constant factor approximation.

Paper proves convergence for private FL on non-Lipschitz convex objectives using normalization instead of clipping.

Improved robustness of machine learning models with controlled Lipschitz constants.

Optimistic method adapted for faster convex-concave min-max problems.

Neural operators learn to solve LQ MFGs efficiently in infinite dimensions.

Square inscribed in a curve made of two graph functions.

ECP optimizes expensive functions without knowing Lipschitz constant.

The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space $\mathbb{H}^{n+1}$. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back, and…

Guarantees uniform convergence for square-root Lipschitz losses.

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…

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.

We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature …