arXiv research

GroupSort neural networks can approximate Lipschitz continuous functions.

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

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…

Investigates Lipschitz continuity in neural networks across various settings.

New Transformers maintain Lipschitz continuity for robustness.

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

Efficient binary sampling method for global optimization of univariate functions with low regret.

Proves conditions for Fourier transforms in rank 1 symmetric spaces.

Efficient algorithm for global optimization of multivariate Lipschitz functions.

ECP optimizes expensive functions without knowing Lipschitz constant.

LALR adapts learning rate for faster convergence in regression and neural nets.

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 studies Lipschitz bounds for integral kernels under differentiability assumptions.

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(Ω, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…

We provide an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear e…

In this paper, we investigate the underlying factor that leads to failure and success in the training of GANs. We study the property of the optimal discriminative function and show that in many GANs, the gradient from the optimal discriminative function is not reliable, which turns out to be the fundamental cause of fa…

Lipschitz RNNs improve stability and performance in various tasks.

New NN design for nonlinear systems control with guarantees.

The curse of dimensionality affects neural network optimization, especially with smooth functions.

New method improves optimization algorithms without Lipschitz smoothness.

In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φ_y+ [φ^{2}/2]_t=w, where w is a bounded function depending o…

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

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

The paper proves Lipschitz continuity of cut times in spacetimes.

BLiE optimizes hyperparameters with theoretical guarantees and superior performance.

CGNNs use wavelets for continuous function generation in infinite-dimensional spaces.

Study examines stability of image-reconstruction algorithms using variational regularization.

Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatur…

The paper proves deep learning can be robust with certain loss functions.

This paper presents a novel nonmyopic adaptive Gaussian process planning (GPP) framework endowed with a general class of Lipschitz continuous reward functions that can unify some active learning/sensing and Bayesian optimization criteria and offer practitioners some flexibility to specify their desired choices for defi…

Proves existence of curved surfaces in hyperbolic space.

Lipschitz maps on metric surfaces are rigid if they preserve area.

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)}(…

We show that for every Lipschitz function $f$ defined on a separable Riemannian manifold $M$ (possibly of infinite dimension), for every continuous $ε:M\to (0,+\infty)$, and for every positive number $r>0$, there exists a $C^\infty$ smooth Lipschitz function $g:M\to\mathbb{R}$ such that $|f(p)-g(p)|\leqε(p)$ for every …

Bayesian optimization and Lipschitz optimization have developed alternative techniques for optimizing black-box functions. They each exploit a different form of prior about the function. In this work, we explore strategies to combine these techniques for better global optimization. In particular, we propose ways to use…

This research explores principles of Lipschitz continuity in neural networks for robustness and generalization.

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

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 $…

Lorentzian distances to Cauchy surfaces fail to be locally equi-Lipschitz.

The paper designs neural networks with assurance for controlling nonlinear systems.

We model how Lipschitz continuity changes during neural network training.

Lipschitz continuity recently becomes popular in generative adversarial networks (GANs). It was observed that the Lipschitz regularized discriminator leads to improved training stability and sample quality. The mainstream implementations of Lipschitz continuity include gradient penalty and spectral normalization. In th…

A new method optimizes in nonstationary environments with many arms efficiently.

We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipsc…

Adversarial attacks against machine learning models are a rather hefty obstacle to our increasing reliance on these models. Due to this, provably robust (certified) machine learning models are a major topic of interest. Lipschitz continuous models present a promising approach to solving this problem. By leveraging the …

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity …

We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact K{ä}hler manifolds. We characterize toric qpsh functions and give necessary and sufficient conditions for them to have finite (weighted) energy, both in terms of the associated convex function in R n , and through t…

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.