arXiv research

Proves planar Lipschitz critical points of area functional are smooth.

Bi-Lipschitz proof for 2-varifolds near critical Allard condition.

We show that any smooth bi-Lipschitz $h$ can be represented exactly as a composition $h_m \circ ... \circ h_1$ of functions $h_1,...,h_m$ that are close to the identity in the sense that each $\left(h_i-\mathrm{Id}\right)$ is Lipschitz, and the Lipschitz constant decreases inversely with the number $m$ of functions com…

Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.

Defines weak geodesics on specific subsets of manifolds.

Accelerated method finds critical points faster on manifolds.

Paper provides estimates for varifolds with critical mean curvature.

The paper studies harmonic graphs in the Heisenberg group and their properties.

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via immersions if the map has no singular points on $M$ in the sense of F.H. Clarke, where $\dim M \leq \dim N$. As its corollary, we have that if a bi-Lipschitz homeomo…

Adaptive LR improves neural network Lipschitz regularity without slowing convergence.

We generalize the notion of integral Menger curvature introduced by Gonzalez and Maddocks by decoupling the powers in the integrand. This leads to a new two-parameter family of knot energies $intM^{p,q}$. We classify finite-energy curves in terms of Sobolev-Slobodeckij spaces. Moreover, restricting to the range of para…

Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.

Efficient local Lipschitz bounds improve neural network robustness.

Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\Cr(u)\equiv \{x:|\nabla u|(x)=0\}$. The results are new even for harmonic functions on $\dR^n$. Given such a $u$, the standard {\it first o…

This research explores principles of Lipschitz continuity in neural networks for robustness and generalization.

Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.

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 Transformers maintain Lipschitz continuity for robustness.

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…

GraN-GAN normalizes gradients for better GAN performance.

The paper proves Lipschitz continuity of cut times in spacetimes.

The paper proposes a method to train NNs with a small Lipschitz constant to improve robustness.

The study shows that certain graphs are regular at boundary points.

Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.

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

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.

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 …

The main result of this paper is: {\bf Theorem.} Let $f:\mathbb{R}^k\rightarrow \mathbb{R}$ be a $C^{1}$ function, so that $\nabla f$ is locally Lipschitz continuous. Assume moreover that $f$ is $C^2$ near its generalised saddle points. Fix real numbers $δ_0>0$ and $0<α<1$. Then there is a smooth function $h:\mathbb{R}…

Novel coordinate descent (CD) methods are proposed for minimizing nonconvex functions consisting of three terms: (i) a continuously differentiable term, (ii) a simple convex term, and (iii) a concave and continuous term. First, by extending randomized CD to nonsmooth nonconvex settings, we develop a coordinate subgradi…

This is the first paper of two ones. Here we prove that two compact Alexandrov surfaces of bounded integral curvature having no peak points are bi-Lipschitz equivalent if they are homeomorphic one to the other. Also conditions under that two ends having finite integral negative curvature are bi-Lipschitz equivalent are…

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

Orthogonium offers unified, efficient layers for robust deep learning.

Study shows prior Lipschitz continuity can improve adversarial robustness of Bayesian Neural Networks.

The paper proves Lipschitz regularity of graph Laplacian eigenvectors on random data clouds.

Study 2D spaces with curvature, finding a graph structure.

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 scalable Lipschitz bounds improve neural network robustness analysis.

While Standard gradient descent is one very popular optimisation method, its convergence cannot be proven beyond the class of functions whose gradient is globally Lipschitz continuous. As such, it is not actually applicable to realistic applications such as Deep Neural Networks. In this paper, we prove that its backtra…

Considering the Teichmüller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point projection to these geodesics is strongly contracting. Consequently, these geodesics are…

This paper reverses a construction by merging boundary critical points into an interior one.

BLiE optimizes hyperparameters with theoretical guarantees and superior performance.

New algorithms for online learning without boundedness or Lipschitz loss assumptions.

The minimal number of critical points is studied for smooth functions on closed manifolds.

We study the Lipschitz metric on Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of Teichmuller space the Lipschitz metric is approximated up to bounded additive distortion by the sup metric on a product of lower-dimensional spaces (similar to the Teichm…

Since their invention, generative adversarial networks (GANs) have become a popular approach for learning to model a distribution of real (unlabeled) data. Convergence problems during training are overcome by Wasserstein GANs which minimize the distance between the model and the empirical distribution in terms of a dif…

For distinct points $p$ and $q$ in a two-dimensional Riemannian manifold, one defines their mediatrix $L_{pq}$ as the set of equidistant points to $p$ and $q$. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes addi…

Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.

The paper trains neural networks with robustness guarantees using semidefinite constraints.