arXiv research

Investigates Lipschitz continuity in neural networks across various settings.

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

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.

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

We address reinforcement learning problems with finite state and action spaces where the underlying MDP has some known structure that could be potentially exploited to minimize the exploration rates of suboptimal (state, action) pairs. For any arbitrary structure, we derive problem-specific regret lower bounds satisfie…

We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg group with its Carnot-Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem.

Proves a new law of robustness for interpolating arbitrary data distributions.

New MIP formulations for neural network Lipschitz constant estimation.

Study shows shallow ReLU networks struggle with high-dimensional Lipschitz functions.

The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.

Study risk-sensitive reinforcement learning with Lipschitz dynamic risk measures, establishing regret bounds.

We prove that any finite dimensional Alexandrov space with a lower curvature bound is locally Lipschitz contractible. As applications, we obtain a sufficient condition for solving the Plateau problem in an Alexandrov space considered by Mese and Zulkowski.

We show that the log-likelihood of several probabilistic graphical models is Lipschitz continuous with respect to the lp-norm of the parameters. We discuss several implications of Lipschitz parametrization. We present an upper bound of the Kullback-Leibler divergence that allows understanding methods that penalize the …

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

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

We study continuous maps between differential manifolds from a microlocal point of view. In particular, we characterize the Lipschitz continuity of these maps in terms of the microsupport of the constant sheaf on their graph. Furthermore, we give lower and upper bounds on the microsupport of the graph of a continuous m…

ECPv2 optimizes Lipschitz functions efficiently and scalably.

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

The paper classifies capillary graphs on manifolds with Ricci lower bounds.

Graph-based framework for provably robust adversarial training.

In this work we compute lower Lipschitz bounds of $\ell_p$ pooling operators for $p=1, 2, \infty$ as well as $\ell_p$ pooling operators preceded by half-rectification layers. These give sufficient conditions for the design of invertible neural network layers. Numerical experiments on MNIST and image patches confirm tha…

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

Let $X$ and $Y$ be length metric spaces. Let $\mathcal H^n$ denote the $n$-dimensional Hausdorff measure. The Lipschitz-Volume Rigidity is a property that if there exists a 1-Lipschitz map $f\colon X\to Y$ and $0<\mathcal H^n(X)=\mathcal H^n(f(X))<\infty$, then $f$ preserves the length of path. This property holds for …

The paper confirms a conjecture about manifolds with positive curvature.

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

New algorithms for online learning without boundedness or Lipschitz loss assumptions.

New algorithm tackles delayed feedback in Lipschitz bandits with sublinear regret.

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

Efficient algorithm for global optimization of multivariate Lipschitz functions.

The paper studies the distance from calibration in sequential prediction, proving upper and lower bounds.

We study the Lipschitz metric on Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of Teichmuller space the Lipschitz metric is approximated up to bounded additive distortion by the sup metric on a product of lower-dimensional spaces (similar to the Teichm…

Proves depth 2 neural networks can't approximate certain functions as well as depth 3 networks.

Proposes a method for obtaining interval bounds in off-policy evaluation.

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

Optimizes privacy-preserving optimization for heavy-tailed data.

New algorithm reduces pricing error by a factor of T^2/3.

Study invariant Lipschitz bandits, improving regret bounds.

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

In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative $1$-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concen…

New law establishes robustness for neural networks with bounded weights.

Verifying correctness of deep neural networks (DNNs) is challenging. We study a generic reachability problem for feed-forward DNNs which, for a given set of inputs to the network and a Lipschitz-continuous function over its outputs, computes the lower and upper bound on the function values. Because the network and the …

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

New lower bound shows bandit convex optimization is harder than previously thought.

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${…

Hyperbolic space outperforms Euclidean in learning hierarchical data.

Bounds on geodesic distances on Stiefel manifold derived from new metrics.

New bounds on SGD's final iterate convergence rate in constant dimension.

Certified algorithms optimize functions with varying costs, providing error bounds.