RVFL networks can efficiently approximate Lipschitz functions in L∞ norm.
problem Efficiently approximating Lipschitz continuous functions in L∞ norm.
method Random Vector Functional Link (RVFL) network with ReLU activation functions, proving approximation in L∞ norm.
result An RVFL with ReLU activation functions can approximate Lipschitz continuous functions in L∞ norm.
A link of an isolated singularity of a two-dimensional semialgebraic surface in R4 is a knot (or a link) in S3. Thus the ambient Lipschitz classification of surface singularities in R4 can be interpreted as a bi-Lipschitz refinement of the topological classification of knots (or links) in S3. We show that, …
The paper extends a link criterion for Lipschitz normal embeddings to definable sets in o-minimal structures.
problem Characterizing Lipschitz normal embeddings of definable sets.
method Extending a known result about subanalytic germs to definable germs in any o-minimal structure.
result The link criterion holds for definable germs in o-minimal structures, but is not sufficient for all homomorphisms.
Efficiently learns Single-Index Models with constant factor approximation.
problem Learning Single-Index Models under L22 loss with unknown link functions. method An efficient algorithm using alignment sharpness for optimization.
result Achieves constant factor approximation to optimal loss for various distributions and link functions.
Study links between surface germs and knot theory in 4D.
problem Understanding the relationship between surface germs and knot theory in R4. method Constructing surface germs XK linked to knots K in S3 and studying their Lipschitz geometry. result Ambient bi-Lipschitz equivalence of surface germs is related to isotopy of knots, and Jones polynomial can recognize non-equivalent germs.
Maps and measures on surfaces link best Lipschitz and least gradient functions.
problem Analyzing maps between surfaces and their geometric properties.
method Duality between best Lipschitz and least gradient maps, geodesic laminations, and transverse measures.
result The infinity harmonic map defines a geodesic lamination and the least gradient map defines a transverse measure.
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.
problem Lack of standard cash-additive methods in Lipschitz spaces.
method Proposes Lipschitz-free space, uses additivity along benchmark-deviation instruments.
result Derives dual representations for convex and coherent risk measures.
Paper studies geometric and combinatorial properties of circular snakes.
problem Exploring geometric and combinatorial properties of circular snakes.
method Definition and investigation of outer Lipschitz geometry, decomposition of Valette link, construction of combinatorial objects, weakly outer Lipschitz classification.
result Existence of canonical decomposition and necessary/sufficient criteria for removing segments or Hölder triangles.
The free factor complex of rank 4+ fails a combinatorial isoperimetric inequality.
problem Failure of combinatorial isoperimetric inequality in the free factor complex.
method Construction of a coarsely Lipschitz function from the upward link of a free factor to integers.
result A loop in the free factor complex requires linearly growing number of 2-simplices to fill.
Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.
problem Classify bi-Lipschitz equivalence of mixed polynomials with inner non-degeneracy.
method Defined metric links and introduced new data to determine bi-Lipschitz equivalence.
result Neither Newton boundary nor C-face diagram is an invariant for bi-Lipschitz equivalence.
Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.
problem Understanding intrinsic Hopf-Lax semigroup and its relation to intrinsic slope.
method Introduces and proves the link between intrinsic Hopf-Lax semigroup and intrinsic slope.
result Intrinsic Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equality.
Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.
problem Characterizing neural network approximations to Lipschitz functions.
method Analyzing L-Lipschitz neural networks and their density in L-Lipschitz functions. result One layer neural networks are dense in the set of all L-Lipschitz functions. We provide bi-Lipschitz invariants for finitely determined map germs f:(Kn,0)→(Kp,0), where K=R or 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.
problem Learning optimal predictors for various loss functions.
method Sharp analysis of Isotron algorithm for agnostic learning.
result Improved sample complexity to ≈ε−2 for bi-Lipschitz link functions. 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.
problem Creating a Lipschitz map with specific local Lipschitz constants on a subset of a manifold.
method Constructs a Lipschitz map that matches a given map on a subset and has a local Lipschitz constant defined by a continuous function.
result A Lipschitz map can be constructed with a local Lipschitz constant prescribed by a continuous function.
Bi-Lipschitz flows approximate a wide range of distributions.
problem Characterizing the expressivity of bi-Lipschitz normalizing flows.
method Linking score regularity to transport map bi-Lipschitzness via probability flow ODE.
result Gaussian pullbacks induced by bi-Lipschitz variance-preserving transport maps are L1-dense among all probability densities. The paper examines bi-Lipschitz triviality of function germs on singular varieties.
problem Analyzing the bi-Lipschitz triviality of deformations of function germs on singular varieties.
method Introducing strongly rational RX-bi-Lipschitz trivial families and providing an infinitesimal criterion for bi-Lipschitz triviality. result Bi-Lipschitz triviality of deformations of f on (X,0) when X and f are homogeneous of the same degree. Paper presents an efficient algorithm for estimating Lipschitz functions from noisy data.
problem Estimating unknown Lipschitz functions from noisy observations.
method Extends max-affine methods to Lipschitz setting using nonlinear feature expansion and adaptive partitioning.
result Achieves minimax convergence rate with respect to intrinsic dimension, up to logarithmic factors.
New approach to certifiably robust neural networks using Boolean function perspective.
problem Lack of principled understanding and certified robustness for ℓ∞ perturbations. method New perspective on Boolean functions, deriving impossibility results, and developing a unified Lipschitz network.
result Unified Lipschitz network that bypasses expressive power limitations and achieves better certified robustness.
GroupSort neural networks can approximate Lipschitz continuous functions.
problem Understanding and improving the expressive power of neural networks with Lipschitz constraints.
method Introduced and studied GroupSort neural networks with constraints on weights, proving their ability to approximate Lipschitz continuous functions.
result GroupSort networks can represent any Lipschitz continuous piecewise linear functions and are well-suited for approximating general 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.
problem Optimizing expensive, non-convex functions with unknown Lipschitz constants.
method ECP minimizes evaluations by focusing on potentially optimal regions, eliminating Lipschitz constant estimation.
result Guaranteed no-regret performance and minimax-optimal regret bounds.
We compute the local Lipschitz constant of ReLU networks precisely.
problem Estimating the local Lipschitz constant of ReLU networks is hard.
method We use a novel approach involving the generalized Jacobian and backpropagation.
result We provide an algorithm to compute the exact Lipschitz constant of ReLU networks.
Extends Lipschitz functions while preserving local constants.
problem Extending Lipschitz functions on metric spaces while maintaining local constants.
method Extends Lipschitz functions on metric spaces while locally preserving the asymptotic Lipschitz constant.
result Sobolev spaces on metric measure spaces are invariant under isomorphism of mm-structures.
Study non-local isoperimetric energies on spheres using a Riemannian autocorrelation function.
problem Analyse non-local isoperimetric energies on spheres.
method Introduce Riemannian autocorrelation function and use it to reformulate and compute the energies.
result Show that the non-local isoperimetric energies can be reformulated and computed using the Riemannian autocorrelation function.
The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.
problem The expressivity of bi-Lipschitz Normalizing Flows in approximating specific target distributions.
method Characterization of expressivity through lower bounds on Total Variation distance and discussion of potential remedies.
result Several target distributions are difficult to approximate using bi-Lipschitz Normalizing Flows, and lower bounds on their approximation are provided.
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 dimM≥dimN, 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.
problem Uniform convergence guarantees for square-root Lipschitz losses.
method Using Rademacher complexity and square root of scalar loss function Lipschitz constant.
result Generalizes previous results and handles non-smooth loss functions.
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.
problem Global optimization of univariate loss functions.
method Binary sampling approach to circumvent hard-to-determine query points in traditional methods.
result At most Llog(3T) and 2.25H regret for L-Lipschitz continuous and H-Lipschitz smooth functions respectively. Extends involutivity to non-Lipschitz subbundles and proves the Frobenius Theorem.
problem Defining involutivity for non-Lipschitz subbundles and proving the Frobenius Theorem.
method Using generalized functions, the Frobenius Theorem is extended to log-Lipschitz subbundles with sharp regularity estimates.
result For log-Lipschitz involutive subbundles, there exists a homeomorphism with specific regularity properties.
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→R can be uniformly approximated by a Lipschitz, C1-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.
problem Understanding the Lipschitz behavior of neural networks.
method Empirical investigation of Lipschitz bounds in different neural network architectures and datasets.
result Remarkable fidelity of the lower Lipschitz bound and a Double Descent trend in both upper and lower bounds.
Proves planar Lipschitz critical points of area functional are smooth.
problem Lawson-Osserman conjecture about smoothness of critical points.
method Outer variations to prove smoothness of critical points.
result Proves conjecture for planar case.
Muon fails to converge on convex Lipschitz functions.
problem Understanding the convergence of Muon on convex and Lipschitz functions.
method Analyzing Muon's performance on convex and Lipschitz functions without error feedback.
result Muon does not converge on convex and Lipschitz functions, regardless of learning rate schedule.
The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.
problem Uniform Lipschitz bounds on geometric functions in Teichmüller space.
method Injectivity radius analysis and Lipschitz bounds on systole function.
result Uniform Lipschitz constant for the square root of the systole function on 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.
problem Estimating the residual Monge-Ampère mass of plurisubharmonic functions.
method General decomposition formula under Sasakian structure, L1-apriori estimate, upper-bound estimate on residual mass. result Upper-bound estimate on residual mass for uniformly directional Lipschitz continuity confirmed.
The paper calculates upper bounds on ReLU network Lipschitz constants.
problem Determining the maximum perturbation size for robustness of neural networks.
method Analyzing ReLU, affine-ReLU, and max pooling functions; combining results; tracking zero elements; using a computational approach.
result The method produces the largest known bounds on minimum adversarial perturbations for large networks.
Study the landscape of Lipschitz functions between manifolds using persistent homology.
problem Understanding the structure of homotopy paths between maps with high Lipschitz constants.
method Using persistent homology to analyze the landscape of Lipschitz functions between manifolds.
result First results on the persistence of higher-dimensional cycles in function spaces.
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…
The Piyavskii-Shubert algorithm is analyzed for global optimization of Lipschitz functions.
problem Maximizing a non-concave Lipschitz function over a compact domain.
method Sequential function evaluations using a bandit-optimization approach.
result New bounds on the number of evaluations needed for optimization accuracy.
LALR adapts learning rate for faster convergence in regression and neural nets.
problem Finding optimal learning rates for faster convergence in regression and neural networks.
method Lipschitz continuity theory applied to Mean Absolute Error and Quantile loss functions.
result Adaptive learning rate policy enables up to 20x faster convergence.