arXiv research

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

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

The paper examines bi-Lipschitz triviality of function germs on singular varieties.

Paper presents an efficient algorithm for estimating Lipschitz functions from noisy data.

New approach to certifiably robust neural networks using Boolean function perspective.

GroupSort neural networks can approximate Lipschitz continuous functions.

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…

ECP optimizes expensive functions without knowing Lipschitz constant.

Extends Lipschitz functions while preserving local constants.

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

We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold $M$ to a connected compact Riemannian manifold $N$, where $\dim M \geq \dim N$, has no singular points on $M$ in the sense of F.H. Clarke, then the map admits a smooth approximation via Ehresmann fibrations. We also show the Reeb s…

Guarantees uniform convergence for square-root Lipschitz losses.

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…

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

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

The local Lipschitz constant of a neural network is a useful metric with applications in robustness, generalization, and fairness evaluation. We provide novel analytic results relating the local Lipschitz constant of nonsmooth vector-valued functions to a maximization over the norm of the generalized Jacobian. We prese…

Training neural networks under a strict Lipschitz constraint is useful for provable adversarial robustness, generalization bounds, interpretable gradients, and Wasserstein distance estimation. By the composition property of Lipschitz functions, it suffices to ensure that each individual affine transformation or nonline…

Let us consider a Riemannian manifold $M$ (either separable or non-separable). We prove that, for every $ε>0$, every Lipschitz function $f:M\rightarrow\mathbb R$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le \Lip(f)+ε$. As a consequence, every Riemannian manifold is uniformly …

Investigates Lipschitz continuity in neural networks across various settings.

Proves planar Lipschitz critical points of area functional are smooth.

Muon fails to converge on convex Lipschitz functions.

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

We consider the finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk functions are the summations of independent and identically distributed (iid) losses. T…

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

The paper calculates upper bounds on ReLU network Lipschitz constants.

Study the landscape of Lipschitz functions between manifolds using persistent homology.

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…

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

Efficient algorithm for global optimization of multivariate Lipschitz functions.

Square inscribed in a curve made of two graph functions.

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

The paper calculates bounds on the local Lipschitz constants of neural network layers.

Paper proves convergence for private FL on non-Lipschitz convex objectives using normalization instead of clipping.

In this paper, we study the convergence of generative adversarial networks (GANs) from the perspective of the informativeness of the gradient of the optimal discriminative function. We show that GANs without restriction on the discriminative function space commonly suffer from the problem that the gradient produced by …

GD-trained shallow ReLU nets learn Lipschitz functions with noise.

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

JacNet learns Jacobians to enforce structure on derivatives for invertibility and Lipschitz functions.

A Lipschitz hypersurface is a hypersurface which locally is the graph of a Lipschitz function. A Lipschitz (or C^1) hypersurface is said to be Levi-flat if it is locally foliated by complex manifolds of complex dimension (n-1). We shall prove that there exist no Lipschitz Levi-flat hypersurfaces in CP^n with n >= 3. Ou…

Enhances SSL with mixup and Lipschitz regularization.

New parameterization of neural networks with Lipschitz bounds for robustness.

Efficient local Lipschitz bounds improve neural network robustness.

We focus our attention on the notion of intrinsic Lipschitz graphs, inside a special class of metric spaces i.e. the Carnot groups. More precisely, we provide a characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2 in terms of their intrinsic distributional gradients.

Lipschitz RNNs improve stability and performance in various tasks.

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

Proves continuum limits of Lipschitz learning using Γ-convergence.

ECPv2 optimizes Lipschitz functions efficiently and scalably.

The study proves surfaces in a specific Heisenberg group must be simple planes.

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