arXiv research

Novel algorithm speeds up computation of Sobolev IPM for graph-based probability measures.

Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.

We prove the developability and $C^{1,1/2}$ regularity of $W^{2,2}$ isometric immersions of $n$-dimensional domains into $R^{n+1}$. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the $W^{2,2}$ strong norm, provided the domain is $C^1$ and convex. Both results fail to …

Let $F : H^q \to H^q$ be a $C^k$-map between Sobolev spaces, either on $\mathbb R^d$ or on a compact manifold. We show that equivariance of $F$ under the diffeomorphism group allows to trade regularity of $F$ as a nonlinear map for regularity in the image space: for $0 \leq l \leq k$, the map $F: H^{q+l} \to H^{q+l}$ i…

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping p…

Study continuity of complex Sobolev functions, with applications to Kaehler metrics.

Develops topological concepts for Morrey-Sobolev bundles in high dimensions.

Paper tackles functional linear regression using spectral algorithms with discrete observations.

Solves Yamabe problem for 3D metrics of Sobolev class $W^{2,q}$.

The paper discusses fractional Sobolev immersions of flat domains into 3D space.

Solves Yamabe problem for Sobolev-class asymptotically hyperbolic manifolds.

We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extreme…

Sharp Sobolev theory for scalar elliptic equations on minimal regular manifolds.

On any complete Riemannian manifold $M$ and for all $p\in [2,\infty)$, we prove a family of second order $L^{p}$-interpolation inequalities that arise from the following simple $L^{p}$-estimate valid for every $u \in C^{\infty}(M)$: $\|\nabla u\|_{p}^p \leq \|u Δ_{p} u\|_1\in [0,\infty],$ where $Δ_p$ denotes the $p…

Paper introduces a new regularization method for kernel gradient descent learning.

Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.

Researchers find second-order estimates for $p$-Laplacian in RCD spaces.

Study on Brownian motion on discrete curve spaces, proving stochastic completeness.

Study on smoothness of solutions to nonlinear equations on Riemannian manifolds.

We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure $μ$. We show that the Sobolev IPM compares two distributions in hig…

Study on mappings in Carnot groups, proving rigidity results.

Sharp bounds for approximating Sobolev functions by ridge functions and networks.

Variational methods for revealing visual concepts learned by convolutional neural networks have gained significant attention during the last years. Being based on noisy gradients obtained via back-propagation such methods require the application of regularization strategies. We present a mathematical framework unifying…

Error estimates for nonlinear PDEs using kernel/GP methods.

The study shows how to regularize weakly harmonic maps using Sobolev norms and Coulomb frames.

New method estimates densities using Sobolev regularization, outperforming existing algorithms.

We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order $n\geq 2$ are metrically complete on the space $\mathcal I^n(S^1,\mathbb R^d)$ of Sobolev immersions of the same regularity and that any two curves i…

The paper proves isometric embedding equations in low Sobolev regularity.

New metrics on curve spaces improve shape analysis.

The transition maps for a Sobolev $G$-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle $P$ is equipped with a Sobolev connection $A$, then one can associate a topological isomorphism class to the pair $\left( P, A\…

Develops Poisson structures on weak Sobolev loop spaces for integrable systems.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at i…

New ADM mass definition for weakly regular manifolds.

Adapts to estimate functions from noisy ERT data.

We show in this note that the Sobolev Discrepancy introduced in Mroueh et al in the context of generative adversarial networks, is actually the weighted negative Sobolev norm $||.||_{\dot{H}^{-1}(ν_q)}$, that is known to linearize the Wasserstein $W_2$ distance and plays a fundamental role in the dynamic formulation of…

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

We study a simplification of GAN training: the problem of transporting particles from a source to a target distribution. Starting from the Sobolev GAN critic, part of the gradient regularized GAN family, we show a strong relation with Optimal Transport (OT). Specifically with the less popular dynamic formulation of OT …

Study on isoperimetric inequalities and regularity of $A$-harmonic functions on surfaces.

Study shows limitations of Lie bracket commutation for nonsmooth vector fields.

We prove the smoothness of the L^2-analytic torsion form on some fiber bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev space, and proving that the Novikov-Shubin invariant remains positive in the Sobolev setti…

We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted Hölder and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy prob…

Study shows harmful overfitting in Sobolev spaces even as training data grows.

This paper compares deeper and wider neural networks for optimal generalization error in Sobolev losses.

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spac…

Sharp log-Sobolev inequalities proved for ${\sf CD}(0,N)$ spaces.

The paper proves a decomposition theorem for forms on sub-Riemannian contact manifolds.

Using simple facts from harmonic analysis, namely Bernstein inequality and Plansherel isometry, we prove that the pseudodifferential equation $Δ^αu+Vu=0$ improves the Sobolev regularity of solutions provided the potential $V$ is integrable with the critical power $n/2α>1$.

Study generalizes Möbius energy to non-smooth sets in arbitrary dimensions.