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. 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.
Maps between certain Lipschitz manifolds are isometries if they preserve volume.
problem Volume preservation and isometry conditions for Lipschitz manifolds.
method Volume-preserving 1-Lipschitz maps from integral currents onto infinitesimally Euclidean Lipschitz manifolds.
result Volume-preserving maps are isometries under given conditions.
Abstract: Lipschitz homeomorphisms are deformed using Perelman's methods.
problem Deformation of Lipschitz homeomorphisms
method Lipschitz analogues of Siebenmann's and Perelman's homeomorphism theory
result Lipschitz stability theorem and gluing theorem
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.
Novikov conjecture reduced to Lipschitz cohomology of groups.
problem Novikov higher signature conjecture for groups.
method Introducing Lipschitz cohomology classes and reducing the conjecture.
result Reduced Novikov conjecture to Lipschitz cohomology.
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…
New MIP formulations for neural network Lipschitz constant estimation.
problem Ensuring robustness of neural networks by calculating their Lipschitz constant.
method Reformulating the neural network Lipschitz estimation problem as a Quadratically Constrained MIP (MIQCQP) problem.
result Solutions of the MIQCQP formulations provide bounds on the Lipschitz constant, with conditions for exactness.
New method for differentially private optimization with general Lipschitz conditions.
problem Differentially private optimization under general Lipschitz conditions.
method Generalized Lipschitz condition for per-sample gradients, tuning clip norm based on minimum per-sample Lipschitz constant.
result Efficacy of the recommended clip norm tuning method verified on 8 datasets.
The paper studies properties of intrinsically Lipschitz constants in metric spaces.
problem Investigating properties of intrinsically Lipschitz constants.
method Introduced Leibniz and product formulas for intrinsic slope.
result Formulated Leibniz and product formulas for intrinsic slope.
New scalable Lipschitz bounds improve neural network robustness analysis.
problem Computing tight Lipschitz bounds for deep neural networks is challenging and computationally expensive.
method Derived new closed-form Lipschitz bounds using more general feasible points of LipSDP, avoiding SDP solvers.
result Improved scalability and precision of Lipschitz estimation for large neural networks.
The paper studies Lipschitz bounds for integral kernels under differentiability assumptions.
problem Understanding the Lipschitz continuity of feature maps associated with integral kernels.
method Analyzes differentiability assumptions to derive explicit formulas for Lipschitz constants and conditions for non-Lipschitz continuity.
result Explicit formulas and conditions for Lipschitz continuity of feature maps associated with various kernels.
We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Ru…
Geodesically complete spaces with curvature bounded above have maps with finite energy that are Lipschitz.
problem Analyzing the properties of geodesically complete spaces with curvature constraints.
method Geometric perturbations of geodesics to curves with zero length on singular sets.
result Every Sobolev map in W1,∞ space has a Lipschitz representative with the same Lipschitz constant as its infinity energy. We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S^n,Y) are not dense in the Sobolev space W^{1,n}(S^n,Y). On the other hand we show that if a metric space Y is Lipschitz (n-1)-connected, then Lipschitz …
Proves rigidity for maps between manifolds using degree theory and current developments.
problem Lipschitz-volume rigidity for maps between metric manifolds and Riemannian manifolds.
method Degree theory and recent developments of Lipschitz-volume rigidity for integral currents.
result Proves a Lipschitz-volume rigidity result for 1-Lipschitz maps.
Paper investigates Lipschitz constants of self-attention modules in neural networks.
problem Lipschitz constants of self-attention modules in neural networks.
method Proved standard dot-product self-attention is not Lipschitz for unbounded input domain. Proposed L2 self-attention that is Lipschitz. Derived upper bound on L2 self-attention's Lipschitz constant.
result Proved standard self-attention is not Lipschitz for unbounded input domain and proposed an alternative L2 self-attention that is Lipschitz.
The thesis defines and proves invariants for manifolds of bounded geometry.
problem Lipschitz-homotopy invariants for manifolds of bounded geometry.
method Definition of a controvariant functor and invariance of the Roe index and ρ-class.
result Lipschitz-homotopy invariants are defined and proven for manifolds of bounded geometry.
Defines a new modulus for Lipschitz surfaces and proves a homological duality theorem.
problem Lipschitz homology classes and their moduli.
method Defining a new modulus dModp and proving a homological duality theorem. result Every relative Lipschitz k-homology class has a unique dual class satisfying a specific modulus product equality. 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.
New Transformers maintain Lipschitz continuity for robustness.
problem Ensuring robustness in Transformers for safety-sensitive applications.
method Introducing gradient-descent-type in-context Transformers with explicit Euler steps of negative gradient flows.
result Universal approximation theorem for Lipschitz continuous Transformers.
The paper extends a measure preserving property to bi-Lipschitz maps between Moran sets.
problem Extending the measure preserving property to Moran sets.
method Analyzing bi-Lipschitz maps between Moran sets.
result Bi-Lipschitz maps between Moran sets preserve measure locally.
Study maps in semidirect products of groups, proving Lipschitz properties without intrinsic dilations.
problem Proving Lipschitz conditions in semidirect products of groups without intrinsic dilations.
method Using equivalent conditions and properties of projection maps in metric spaces.
result Proves the same Lipschitz results as in Carnot groups, without intrinsic dilations.
New metric measure space theory for Lipschitz constants.
problem Defining and characterizing Cheeger energy in metric measure spaces.
method Adapting Cheeger theory to intrinsically Lipschitz sections.
result Characterization of intrinsic Cheeger energy in terms of relaxed slope.
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.
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.
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.
The paper proves Lipschitz continuity of cut times in spacetimes.
problem Lipschitz continuity of cut times in globally hyperbolic spacetimes.
method Adapted Itoh-Tanaka method to Lorentzian setting.
result Lipschitz continuity of cut times with quantitative estimates.
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.
Generative adversarial networks (GANs) are one of the most popular approaches when it comes to training generative models, among which variants of Wasserstein GANs are considered superior to the standard GAN formulation in terms of learning stability and sample quality. However, Wasserstein GANs require the critic to b…
New method tightens Lipschitz bounds for CNNs efficiently.
problem Lipschitz regularization of Convolutional Neural Networks (CNNs).
method Using Toeplitz matrix theory, introduces a tight and computationally efficient upper bound for convolutional layers.
result Developed an algorithm to train Lipschitz regularized CNNs.
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.
The paper proves Rademacher's theorem for Heisenberg groups.
problem Proving Rademacher's theorem for Heisenberg groups.
method New definition of intrinsic Lipschitz graphs, extension and approximation theorems, use of Heisenberg currents and Rumin's complex.
result Rademacher's theorem for intrinsic Lipschitz graphs in Heisenberg groups.
Study quantifies convergence of Alexandrov spaces without collapsing.
problem Quantifying convergence of Alexandrov spaces without collapsing.
method Lipschitz homotopy convergence for Alexandrov spaces.
result Lipschitz homotopies can be chosen to preserve singular strata.
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…
In this paper we describe the notion of a weak lipschitzianity of a mapping on a Cq stratification. We also distinguish a class of regularity conditions that are in some sense invariant under definable, locally Lipschitz and weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B) condition and the …
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 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…
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:H→Rm is injective, with (α(x))k=∣<x,fk>∣2, where $…
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. 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.
Lipschitz and horizontal maps from an n-dimensional space into the (2n+1)-dimensional Heisenberg group $\H^n$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Hajłasz-Lukyanenko-Tyson constructed horizontal maps from Sk to $\H^n$ which factor through n-spheres and sh…
Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.
problem Extending diffeomorphisms to global mappings of manifolds.
method Elementary argument for diffeomorphisms, deep results for homeomorphisms and bi-Lipschitz mappings.
result Extension of Palais' result to homeomorphisms and bi-Lipschitz mappings.
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 study examines Lipschitz normally embedded Hölder triangles in 4D space.
problem Comparing ambient and outer Lipschitz geometry of Hölder triangles.
method Analyzes Lipschitz normally embedded Hölder triangles in \(\mathbb{R}^4\).
result Infinitely many equivalence classes of microknots.
Bi-Lipschitz proof for 2-varifolds near critical Allard condition.
problem Proving bi-Lipschitz homeomorphism for 2-varifolds near critical Allard condition.
method Analyzing 2-varifolds with critical Allard condition and small mean curvature.
result 2-varifold is bi-Lipschitz homeomorphic to a flat disk.
Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.
problem Characterize the Lipschitz homotopy groups of contact 3-manifolds.
method Sub-Riemannian geometry, geometric measure theory, biLipschitz equivalence, purely unrectifiable sets.
result Contact 3-manifolds are K(π,1) spaces with uncountably generated first homotopy groups. This work focuses on important step in quantitative topology: given homotopic mappings from Sm to Sn of Lipschitz constant L, build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant). The present paper resolves this problem for the first case where Hopf invariant p…