arXiv research

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

A link of an isolated singularity of a two-dimensional semialgebraic surface in $R^4$ is a knot (or a link) in $S^3$. Thus the ambient Lipschitz classification of surface singularities in $R^4$ can be interpreted as a bi-Lipschitz refinement of the topological classification of knots (or links) in $S^3$. We show that, …

The paper extends a link criterion for Lipschitz normal embeddings to definable sets in o-minimal structures.

Efficiently learns Single-Index Models with constant factor approximation.

Study links between surface germs and knot theory in 4D.

Maps and measures on surfaces link best Lipschitz and least gradient functions.

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…

Develops risk measures on Lipschitz spaces for financial positions.

Paper studies geometric and combinatorial properties of circular snakes.

The free factor complex of rank 4+ fails a combinatorial isoperimetric inequality.

Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

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

We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. The aim of the paper is to provide partial answers to the following questions: Does the bi-Lipschitz type of a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R…

New algorithm reduces sample complexity for omnipredictors of SIMs.

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…

Stochastic Lipschitz bandit algorithms balance exploration and exploitation, and have been used for a variety of important task domains. In this paper, we present a framework for Lipschitz bandit methods that adaptively learns partitions of context- and arm-space. Due to this flexibility, the algorithm is able to effic…

The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations…

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

Bi-Lipschitz flows approximate a wide range of distributions.

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.

Study non-local isoperimetric energies on spheres using a Riemannian autocorrelation function.

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.