New method improves optimization algorithms without Lipschitz smoothness.
problem Improving optimization algorithms in the absence of Lipschitz smoothness.
method Dual kernel conditioning (DKC) to provide dual Lipschitz continuity.
result First complexity bounds and iterate convergence for random reshuffling mirror descent.
Stokes theorem holds for Lipschitz forms on a smooth manifold.
The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothn…
Method approximates Lipschitz domains with smoother shapes.
problem Approximating bounded Lipschitz domains.
method Sequence of smooth, bounded domains with weak curvatures.
result Uniform isocapacitary estimates for approximating sets.
Smooth solutions found for a specific type of Yamabe problem.
problem Regularity of viscosity solutions to the σk-Yamabe problem in the negative cone. method Analysis of Lipschitz viscosity solutions with specific assumptions.
result Existence and smoothness of solutions away from a negligible set.
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.
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 …
New shuffling methods improve convergence without Lipschitz smoothness.
problem Lack of convergence guarantees for shuffling methods under non-Lipschitz conditions.
method Revisit shuffling methods, prove convergence under general bounded variance condition.
result Matched current best-known convergence rates without Lipschitz smoothness.
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.
We prove that Lipschitz intrinsic graphs in the Heisenberg groups Hn, with n>1, which are vanishing viscosity solutions of the minimal surface equation are smooth.
The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…
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…
Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
Enhances robustness of deep neural networks with randomized smoothing.
problem Improving robustness of deep neural networks against noisy inputs and adversarial attacks.
method Introduces a variance-margin trade-off approach to increase certified robust radius using pre-trained models.
result Significant improvement in certified accuracy compared to state-of-the-art methods.
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 …
New framework tightens certified robustness gaps in machine learning models.
problem Persistent gap between theoretical certified robustness and empirical accuracy.
method Leverages Lipschitz continuity and novel confidence intervals.
result Improves robust accuracy, compressing the gap between theory and practice.
The study connects polyhedral manifolds to Riemannian ones with geometric bounds.
problem Connecting polyhedral manifolds to Riemannian manifolds with geometric constraints.
method Using a theorem by C. Lange and B. Bowditch, the study bounds the curvature and injectivity radius of Riemannian manifolds.
result Polyhedral manifolds with bounded geometry are bi-Lipschitz homeomorphic to Riemannian manifolds with controlled curvature and injectivity radius.
New bounds for online portfolio selection without smoothness assumptions.
problem Online portfolio selection with non-Lipschitz, non-smooth losses.
method Data-dependent bounds using novel smoothness characterizations and FTRL with self-concordant regularizers.
result Achieves logarithmic regrets when data is 'easy' and sublinear worst-case regrets.
Sparse Polyak improves high-dimensional statistical estimation.
problem High-dimensional statistical estimation problems with growing problem dimension.
method Sparse Polyak modifies Polyak's adaptive step size to estimate restricted Lipschitz smoothness.
result Sparse Polyak achieves optimal statistical precision with fewer iterations.
Corners can be identified by a drum's sound spectrum.
problem Determining the presence of corners in a drum's shape from its sound.
method Proving spectral invariance of corners in domains with Lipschitz, piecewise smooth boundaries.
result Corners are uniquely determined by a drum's spectrum among domains with fixed genus.
We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous ε:M→(0,+∞), and for every positive number r>0, there exists a C∞ smooth Lipschitz function g:M→R such that ∣f(p)−g(p)∣≤ε(p) for every …
Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.
problem Existence time of the Willmore flow for various initial conditions.
method Established minimal existence time for Willmore flow using geometric data and conservation laws.
result Minimal existence time is a function of geometric data for general weak Lipschitz initial data.
We consider the stochastic composition optimization problem proposed in \cite{wang2017stochastic}, which has applications ranging from estimation to statistical and machine learning. We propose the first ADMM-based algorithm named com-SVR-ADMM, and show that com-SVR-ADMM converges linearly for strongly convex and Lipsc…
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. 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 …
Paper proves convergence for private FL on non-Lipschitz convex objectives using normalization instead of clipping.
problem Lack of convergence results for differentially private federated learning with non-Lipschitz objectives.
method Developed a convergence result for private FL on smooth convex objectives without assuming Lipschitzness, using normalization instead of clipping.
result Normalization-based private FL algorithm converges better than clipping-based counterpart on smooth convex functions.
Study Lipschitz regularity for manifold-constrained ROF model on curved surfaces.
problem Lipschitz regularity for manifold-constrained ROF model on curved surfaces.
method Generalization of ROF model, existence and uniqueness of minimizers, regularity results on PDE system.
result Lipschitz regularity of minimizers without convexity requirements.
We prove the following new characterization of Cp (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a Cp smooth (Lipschitz) bump function if and only if it has another Cp smooth (Lipschitz) bump function f such that f′(x)=0 for every point x in the interior of the …
We prove that two-step analytic sub-Riemannian structures on a compact analytic manifold equipped with a smooth measure and Lipschitz Carnot groups satisfy measure contraction properties.
Smoothness analysis of adversarial training reveals L∞ constraints cause more non-smoothness.
problem Non-smoothness of adversarial training loss function.
method Analyzed the smoothness of adversarial training loss function using optimal attacks for model parameters.
result The L∞ constraint causes more non-smoothness than L2 constraint. 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 map between manifolds is an isometry if it's Lipschitz and scalar curvature bounded.
problem Characterizing maps between manifolds based on their scalar curvature and Lipschitz continuity.
method Spectral properties of Dirac operators and index theory for low regularity metrics and bundles.
result A 1-Lipschitz map between manifolds is an isometry if it has bounded scalar curvature.
We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in R2n, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…
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…
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 study examines metrics on Riemannian spaces with bounded properties and finds conditions for Lipschitz and uniform bounds.
problem Investigating bounded rough Riemannian metrics and their properties.
method Analyzing the structure of bounded rough Riemannian metrics and finding conditions for Lipschitz and uniform bounds.
result Weak conditions are identified for Lipschitz and uniform bounds on the metrics.
Smooth functions preserve Zygmund class on curves.
problem Characterizing functions based on their behavior on smooth curves.
method Analyzing functions through their behavior on smooth curves and mappings between Banach spaces.
result Functions in Zygmund class preserve Zygmund class on smooth curves.
Let X and Y be length metric spaces. Let Hn denote the n-dimensional Hausdorff measure. The Lipschitz-Volume Rigidity is a property that if there exists a 1-Lipschitz map f:X→Y and 0<Hn(X)=Hn(f(X))<∞, then f preserves the length of path. This property holds for …
New heat dispersion laws established for smooth compact manifolds.
problem Understanding heat dispersion in smooth compact manifolds.
method Established new heat dispersion laws through Theorem 1.1 and explored them further with Propositions 3.1 and 3.2.
result New heat dispersion laws for smooth compact manifolds.
Analytic functions on specific domains are characterized by their smoothness and composites with polynomial curves.
problem Characterizing real analytic functions on closed subanalytic domains.
method Analyzing functions defined on closed uniformly polynomially cuspidal sets in Rn using composites with polynomial curves. result Conditions for a function to be real analytic are effectively related to the regularity of the boundary of the domain.
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L2 for metrics with finitely differentiable tensor. The study provides optimal estimates for surfaces close to constant mean curvature.
problem Optimizing estimates for surfaces near constant mean curvature.
method Bi-Lipschitz and W2,2 parametrization for surfaces with density close to one and small Willmore energy. result Quantitative rigidity for L2-almost CMC surfaces. 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 rigidity of maps between manifolds with scalar curvature constraints.
problem Lipschitz rigidity problem in scalar curvature geometry.
method Harmonic map heat flow coupled with Ricci flow.
result Continuous maps between manifolds with scalar curvature constraints are either isometries or have scalar curvature strictly less than the sphere.
Enhances SSL with mixup and Lipschitz regularization.
problem Improving SSL performance with limited labeled data.
method Adopts mixup for smoothness, adds adversarial Lipschitz regularization.
result Combined regularization improves SSL performance and robustness.
New method balances multivariate model fitting for mixed likelihoods.
problem Multivariate models often fit only a subset of observed variables.
method Lipschitz standardization for data preprocessing.
result Lipschitz standardization leads to more accurate multivariate models.
Proves Hawking's theorem for less smooth spacetime metrics.
problem Proving Hawking's singularity theorem for less smooth spacetime metrics.
method New estimates for Ricci curvature and a segment-type inequality for volume control.
result Proves Hawking's singularity theorem for Lipschitz metrics.
Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.
problem Characterizing dense subgroups of algebraic groups.
method Bi-Lipschitz rigidity theorem for Zariski dense discrete subgroups.
result No C1-smooth slim limit set for higher rank semisimple algebraic groups.