arXiv research

Study shows limitations of Lie bracket commutation for nonsmooth vector fields.

The paper examines convergence of distances in Lipschitz structures on manifolds.

Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit…

MLDL preserves manifold geometry in vector transformations.

We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.

A generalization of the Flow-box Theorem is given. The assumption of continuous differentiability of the vector field is relaxed to a local Lipschitz condition. The theorem holds in any Banach space.

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

The paper extends Frobenius-type theorems to non-smooth settings with Hölder estimates.

In this paper, we study Lipschitz-Fredholm vector fields on Bounded-Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if $M$ is a connected smooth bounded-Fréchet-Finsler manifold endowed with a strengthened connection $\mathcal{K}$ and if $ξ$ is a smooth Lipschitz-Fr…

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

Study geometric flows with varying parameters and prove continuous dependence.

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

Introduces new geodesic fields for Finsler manifolds.

Develops a new exponential map for time-varying vector fields.

In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the cont…

Flow Matching improves Wasserstein 1 distance convergence in high dimensions.

In this paper we provide an up-to-date survey on the study of Lipschitz equivalence of self-similar sets. Lipschitz equivalence is an important property in fractal geometry because it preserves many key properties of fractal sets. A fundamental result by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, …

We prove that the Hilbert Geometry of a convex set is bi-lipschitz equivalent to a normed vector space if and only if the convex is a polytope.

Study of cosmic microwave background polarization using spin random fields.

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

We prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current; we derive explicit formulae for the boundary, the mass and the first variations a…

We compute the local Lipschitz constant of ReLU networks precisely.

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype {equation*} \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where $p\ge 2$, $ \ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $μ$, …

RVFL networks can efficiently approximate Lipschitz functions in L∞ norm.

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

Scalable method bounds Lipschitz constant of generative models.

Just as an explicit parameterisation of system dynamics by state, i.e., a choice of coordinates, can impede the identification of general structure, so it is too with an explicit parameterisation of system dynamics by control. However, such explicit and fixed parameterisation by control is commonplace in control theory…

SRNF framework extends surface distance to Lipschitz surfaces.

Study on GEPs with generative priors, showing optimal statistical rates and proposing an iterative algorithm.

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

The paper proves Lipschitz continuity of cut times in spacetimes.

The study examines the normal growth exponent of submanifolds in negatively curved manifolds.

Defines a new modulus for Lipschitz surfaces and proves a homological duality theorem.

The paper explores MAE as a loss function for DNN vector-to-vector regression, proving its advantages over MSE.

Second paper in series solves Einstein vacuum equations for three impulsive waves.

Revisits shallow neural networks using Lipschitz norms and measures.

Our work proves robustness of embedding schemes to discrete changes in text.

Paper transforms torse-forming vector fields into simpler forms.

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

Optimistic bounds for multi-output learning using self-bounding Lipschitz condition.

In binary classification and regression problems, it is well understood that Lipschitz continuity and smoothness of the loss function play key roles in governing generalization error bounds for empirical risk minimization algorithms. In this paper, we show how these two properties affect generalization error bounds in …

The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.

The contraction inequality for Rademacher averages is extended to Lipschitz functions with vector-valued domains, and it is also shown that in the bounding expression the Rademacher variables can be replaced by arbitrary iid symmetric and sub-gaussian variables. Example applications are given for multi-category learnin…

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold $M$. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the …

Paper introduces a neural network for consistent estimation of optimal transport maps.

Conformal vector fields on LCP manifolds are orthogonal and Killing.

Standard Transformers approximate Hölder functions and achieve optimal nonparametric regression rate.