arXiv research

The study examines metrics on Riemannian spaces with bounded properties and finds conditions for Lipschitz and uniform bounds.

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

This paper refines homotopy theory for cubical sets and uniform spaces.

New method for differentially private optimization with general Lipschitz conditions.

The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

We prove a uniform extension result for contracting maps defined on subsets of Hadamard manifolds subject to curvature bounds.

Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.

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

Proposes a weaker version of Strong Cosmic Censorship with curvature bounds.

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

Method approximates Lipschitz domains with smoother shapes.

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary codimension for L less than or equal to 1/4.

Guarantees uniform convergence for square-root Lipschitz losses.

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

We show that the isoperimetric profile $h_{g(t)}(ξ)$ of a compact Riemannian manifold $(M,g)$ is jointly continuous when metrics $g(t)$ vary continuously. We also show that, when $M$ is a compact surface and $g(t)$ evolves under normalized Ricci flow, $h^2_{g(t)}(ξ)$ is uniform Lipschitz continuous and hence $h_{g(t)}(…

This paper addresses error bounds and posterior variance for Gaussian process regression.

This paper examines weight initialization for 1-Lipschitz networks to improve robustness against adversarial attacks.

A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over…

Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure and uniform error bounds have been derived, which allow safe control based on thes…

The paper introduces new functors for cohomology groups of manifolds.

Paper introduces risk assessment for contextual bandits without experiments.

Unified framework for uniform signal recovery in nonlinear GCS with 1-bit/quantized measurements.

Study on residual Monge-Ampère mass of complex functions with directional Lipschitz continuity.

We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm that induces d. There is an Out(F_n)-invariant potential Ψon Outer space such that when the Lipschitz norm is corrected by the derivative of Ψ, the resulting norm is quasisymmetric. As an application, we give new …

New bounds for SGLD show error decreases with more data.

New sampling algorithms for complex distributions without log-concavity.

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…

Counterexamples show failure of uniform laws of large numbers for subdifferentials.

New bound relaxes uniform gradient norm assumptions for PAC-Bayesian bounds.

New sampling bounds improve uniform coverage verification in machine learning.

The paper proves uniqueness of evolving graphs by mean curvature flow under specific conditions.

New stability bounds for SGD on nonsmooth convex losses.

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we provide a technique which transfers results on uniform approximation on bounded …

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

Algorithm-dependent generalization error bounds are central to statistical learning theory. A learning algorithm may use a large hypothesis space, but the limited number of iterations controls its model capacity and generalization error. The impacts of stochastic gradient methods on generalization error for non-convex …

New research shows many batch selection methods for training work just as well as full batch training.

New approach achieves optimal rates for differentially private stochastic convex optimization with heavy-tailed gradients.

Develops a framework for distilling flow models from few steps.

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

Unified framework for risk-aware policy learning in contextual bandits.

The study connects Kato bounds to finite-dimensional RCD spaces.

Revisits shallow neural networks using Lipschitz norms and measures.

Develops a new framework for temporal anchoring in deep embedding spaces.

We study controlled systems which are uniformly observable and differentially observable with an order larger than the system state dimension. We establish that they may be transformed into a (partial) triangular canonical form but with possibly non locally Lipschitz functions. We characterize the points where this Lip…

Uniform covers with a finite-dimensional nerve are rare (i.e., do not form a cofinal family) in many separable metric spaces of interest. To get hold on uniform homotopy properties of these spaces, a reasonably behaved notion of an infinite-dimensional metric polyhedron is needed; a specific list of desired properties …

New algorithm improves online learning with reduced discretization.

New scalable Lipschitz bounds improve neural network robustness analysis.