arXiv research

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

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

We give some new methods, based on Lipschitz extension theorems, for bounding filling invariants of subsets of nonpositively curved spaces. We apply our methods to find sharp bounds on higher-order Dehn functions of Sol_{2n+1}, horospheres in euclidean buildings, Hilbert modular groups, and certain S-arithmetic groups.

The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.

The study establishes minimax bounds for estimating operators from noisy samples.

Sharp 2-Wasserstein bounds for DDPMs derived from Föllmer process.

Optimization in the presence of sharp (non-Lipschitz), unpredictable (w.r.t. time and amount) changes is a challenging and largely unexplored problem of great significance. We consider the class of piecewise Lipschitz functions, which is the most general online setting considered in the literature for the problem, and …

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…

Efficiently learns Single-Index Models with constant factor approximation.

We provide sharp empirical estimates of expectation, variance and normal approximation for a class of statistics whose variation in any argument does not change too much when another argument is modified. Examples of such weak interactions are furnished by U- and V-statistics, Lipschitz L-statistics and various error f…

New bounds for neural networks ensure robustness and accuracy.

Full-batch GD achieves generalization close to any stationary point with fewer assumptions.

Upper bounds on nullhomotopy volumes in nilpotent spaces are refined.

The paper proves inequalities linking geometric norms and Thurston norms in hyperbolic 3-manifolds.

Nonparametric Thompson Sampling achieves optimal regret for risk-averse bandits with sub-Gaussian rewards.

Establish optimal Lipschitz lower bounds for functions on manifolds with negative curvature, revealing interplay between width, boundary area, and topology.

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

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

We study the concentration of random kernel matrices around their mean. We derive nonasymptotic exponential concentration inequalities for Lipschitz kernels assuming that the data points are independent draws from a class of multivariate distributions on $\mathbb R^d$, including the strongly log-concave distributions u…

We show that stochastic interpolation flow maps are Lipschitz with a sharp constant.

The paper explores how close two Lipschitz functions can be without their difference exceeding a certain bound.

We show that, for all $α\geq 0$, the generalized Grushin plane $\mathbb{G}_α$ is bi-Lipschitz homeomorphic to a $2$-dimensional quasiplane in the Euclidean space $\mathbb{R}^{[α]+2}$, where $[α]$ is the integer part of $α$. The target dimension is sharp. This generalizes a recent result of Wu.

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…

The study proves a gap theorem for CAT(0) spaces with a constant below 1/(6√π).

New scalable Lipschitz bounds improve neural network robustness analysis.

In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…

We show that there is no bi-Lipschitz homeomorphism of $\mathbb{R}^2$ that maps a spiral with a sub-exponential decay of winding radii to an unwinded arc. This result is sharp as shows an example of a logarithmic spiral.

The paper examines eigenvalues and inequalities on Riemannian manifolds.

The paper calculates upper bounds on ReLU network Lipschitz constants.

We consider the goodness-of-fit testing problem of distinguishing whether the data are drawn from a specified distribution, versus a composite alternative separated from the null in the total variation metric. In the discrete case, we consider goodness-of-fit testing when the null distribution has a possibly growing or…

Geodesically complete spaces with curvature bounded above have maps with finite energy that are Lipschitz.

Investigates Lipschitz continuity in neural networks across various settings.

New parameterization of neural networks with Lipschitz bounds for robustness.

Efficient local Lipschitz bounds improve neural network robustness.

New method for differentially private optimization with general Lipschitz conditions.

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

The thesis defines and proves invariants for manifolds of bounded geometry.

The study connects polyhedral manifolds to Riemannian ones with geometric bounds.

We examine the impact of learning Lipschitz continuous models in the context of model-based reinforcement learning. We provide a novel bound on multi-step prediction error of Lipschitz models where we quantify the error using the Wasserstein metric. We go on to prove an error bound for the value-function estimate arisi…

We provide convergence guarantees in Wasserstein distance for a variety of variance-reduction methods: SAGA Langevin diffusion, SVRG Langevin diffusion and control-variate underdamped Langevin diffusion. We analyze these methods under a uniform set of assumptions on the log-posterior distribution, assuming it to be smo…

The projection of a compact oriented submanifold M^{n-1} in R^{n+1} on a hyperplane P^{n} can fail to bound any region in P. We call this ``projecting to zero.'' Example: The equatorial S^1 in S^2 projects to zero in any plane containing the x_3-axis. Using currents to make this precise, we show: A lipschitz (homology)…

Sharp distance estimates for compact spin manifolds using Dirac operator.

New methods for calculating curvature in graph theory.

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

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.

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

Scalable method bounds Lipschitz constant of generative models.

Improved KL convergence bounds for score diffusion models without restrictive assumptions.