arXiv research

Method approximates Lipschitz domains with smoother shapes.

New Transformers maintain Lipschitz continuity for robustness.

The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.

GroupSort neural networks can approximate Lipschitz continuous functions.

Bi-Lipschitz flows approximate a wide range of distributions.

Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.

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…

Study shows shallow ReLU networks struggle with high-dimensional Lipschitz functions.

Scalable method bounds Lipschitz constant of generative models.

The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.

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 …

ELF simplifies normalizing flows, making them more efficient and universal.

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

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…

The paper simplifies arguments for stationary varifolds results.

Study shows limits on deep and shallow neural networks for approximating compact sets.

Study identifies three quantization regimes for ReLU networks.

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

Neural operators learn to solve LQ MFGs efficiently in infinite dimensions.

Optimizes Lipschitz estimates for partitions of unity and characterizes spaces with Assouad-Nagata dimension.

Residual flows are shown to approximate MMD well.

Adversarial attacks against machine learning models are a rather hefty obstacle to our increasing reliance on these models. Due to this, provably robust (certified) machine learning models are a major topic of interest. Lipschitz continuous models present a promising approach to solving this problem. By leveraging the …

We show that the Kuratowski imbedding of a Riemannian manifold in L^\infty, exploited in Gromov's proof of the systolic inequality for essential manifolds, admits an approximation by a (1+C)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every C>0. Our key tool is the first variation formula thought of…

We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensiona…

Wide deep neural networks with Gaussian weights approximate Gaussian processes closely.

New method tightens Lipschitz bounds for CNNs efficiently.

Existing depth separation results for constant-depth networks essentially show that certain radial functions in $\mathbb{R}^d$, which can be easily approximated with depth $3$ networks, cannot be approximated by depth $2$ networks, even up to constant accuracy, unless their size is exponential in $d$. However, the func…

We derive error estimates for multinomial approximations of American options in a multidimensional jump--diffusion Merton's model. We assume that the payoffs are Markovian and satisfy Lipschitz type conditions. Error estimates for such type of approximations were not obtained before. Our main tool is the strong approxi…

The study establishes equivalence of conditions on metric manifolds with finite volume.

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…

Study proves existence and uniqueness for differential equations with non-Lipschitz coefficients.

Sparse neural networks can match dense models on Lipschitz functions.

We study the robustness of active learning (AL) algorithms against prior misspecification: whether an algorithm achieves similar performance using a perturbed prior as compared to using the true prior. In both the average and worst cases of the maximum coverage setting, we prove that all $α$-approximate algorithms are …

The paper uses neural networks to learn system dynamics from data with Lipschitz regularization.

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 2D spaces with curvature, focusing on structure and approximations.

Deep networks can approximate functions with fewer learnable parameters than previously thought.

ResNets can approximate input distances under certain conditions, but existing theory is flawed.

ECP optimizes expensive functions without knowing Lipschitz constant.

Paper presents an efficient algorithm for estimating Lipschitz functions from noisy data.

Dense neural networks can't approximate all functions.

The paper proves Rademacher's theorem for Heisenberg groups.

New approach to certifiably robust neural networks using Boolean function perspective.

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we provide a technique which transfers results on uniform approximation on bounded …

Proves depth 2 neural networks can't approximate certain functions as well as depth 3 networks.

We show that for every Lipschitz function $f$ defined on a separable Riemannian manifold $M$ (possibly of infinite dimension), for every continuous $ε:M\to (0,+\infty)$, and for every positive number $r>0$, there exists a $C^\infty$ smooth Lipschitz function $g:M\to\mathbb{R}$ such that $|f(p)-g(p)|\leqε(p)$ for every …

We study convergence properties of the full truncation Euler scheme for the Cox-Ingersoll-Ross process in the regime where the boundary point zero is inaccessible. Under some conditions on the model parameters (precisely, when the Feller ratio is greater than three), we establish the strong order 1/2 convergence in $L^…

Counterexamples show failure of uniform laws of large numbers for subdifferentials.