arXiv research

New sampling algorithm for non-smooth potentials.

Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.

Paper proposes an algorithm for sampling from complex mixture distributions without requiring smoothness.

Smoothness proven for conical Calabi-Yau potentials on Fano cones.

We give upper bounds for the Bergman kernels associated to tensor powers of a smooth positive line bundle in terms of the rate of growth of the Taylor coefficients of the Kähler potential. As applications, we obtain improved off-diagonal rate of decay for the classes of analytic, quasi-analytic, and more generally Gevr…

Proves invertibility of layer potentials for generalized Stokes operators on smooth domains.

New potentials found for sheaves on Calabi-Yau 4-folds.

New method approximates sampling from smooth potential distributions using a vanishing penalty.

LMC algorithm achieves efficient sampling from complex distributions with specific tail behaviors.

The paper characterizes potential functions whose level sets are orbits in mechanical systems.

We prove that in smooth Markovian continuous-time economies with potentially complete asset markets, Radner equilibria with endogenously complete markets exist.

The paper describes flat Hessian metrics on surfaces and their potentials.

This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.

This work improves SGMs' convergence guarantees for semiconvex distributions with discontinuous gradients.

We propose a new algorithm---Stochastic Proximal Langevin Algorithm (SPLA)---for sampling from a log concave distribution. Our method is a generalization of the Langevin algorithm to potentials expressed as the sum of one stochastic smooth term and multiple stochastic nonsmooth terms. In each iteration, our splitting t…

Examines discrete curvature's relation to smooth curvature in 3 spaces.

The Yamabe invariant is linked to static potentials and eigenvalues.

Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions. In this paper, we give simple sufficient conditions for a smooth function to be c-convex when the cost is given by minimizing a Lagrangian action.

Some methods based on simple regularizing geometric element transformations have heuristically been shown to give runtime efficient and quality effective smoothing algorithms for meshes. We describe the mathematical framework and a systematic approach to global optimization-based versions of such methods for mixed volu…

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler…

The paper investigates subelliptic harmonic maps with potential using heat flow.

Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $τ$ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete …

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that for the Schrödinger operator $Δ+V$ with potential $V\in C^{1,α}(M_0)$ for some $α>0$, the Dirichlet-to-Neumann map $N|_Γ$ measured on an open set $Γ\subset \partial M_0$ determines uniquely the potential $V$. We also discuss briefly the cor…

The paper proves optimal smoothness for certain Lagrangian graphs with specific Hölder continuity.

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero poten…

Derivatives of sub-Riemannian geodesics are always $L_p$-Hölder continuous.

Using the tractor calculus to study smooth metric measure spaces, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the Weyl tractor $W$ to the setting of smooth metric measure space…

Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is comparable to the distance between the original densities. Therefore, optimal transpo…

Let X be a quasiprojective manifold given by the complement of a divisor $\bD$ with normal crossings in a smooth projective manifold $\bX$. Using a natural compactification of $X$ by a manifold with corners $\tX$, we describe the full asymptotic behavior at infinity of certain complete Kahler metrics of finite volume o…

New methods improve online matrix optimization with reduced computational cost.

LMC algorithm converges to target in Chi-squared and Renyi divergence.

Locally conformally Kahler (LCK) manifolds with potential are those which admit a Kahler covering with a proper, automorphic Kaehler potential. Existence of a potential can be characterized cohomologically as a vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential ar…

We extend the Feynman-Kac formula for Schrödinger type operators on vector bundles over noncompact Riemannian manifolds to possibly very singular potentials that appear in hydrogen like quantum mechanical problems and that need not be bounded from below or locally square integrable. This path integral formula is then u…

Paper improves neural network robustness certification with tighter radii estimates.

We give a sharp upper bound on the vanishing order of solutions to Schrödinger equation, in the case that the potential is of class $\mathcal{C}^1$ on a smooth compact manifold.

In this paper we show that every area minimizing cone C^{n-1} in R^n can be approximated by entirely smooth area minimizing hypersurfaces. This extensively uses hyperbolic unfoldings of such hypersurfaces and the resulting potential theory for the Jacobi field operator. Applications include the splitting theorem in sca…

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that the Cauchy data space (or Dirichlet-to-Neumann map $\mc{N}$) of the Schrödinger operator $Δ+V$ with $V\in C^2(M_0)$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at 0 …

Proposes a differentiable LSE-ICNN for modeling multi-well potentials.

This work is based on the approach developed by J.~Dorfmeister, F.~Pedit and H.~Wu [GANG and KITCS preprint, Report KITCS94-4-1] to construct maps $Φ:D\rightarrow R^3$, $D$ being the unit disk in $C$, whose images are surfaces of constant mean curvature. They start from certain meromorphic one forms, so called meromorp…

New algorithm extends LMC to more complex potentials.

This paper explores how entropic regularization improves Wasserstein estimators' performance.

Diffuse interface methods have recently been introduced for the task of semi-supervised learning. The underlying model is well-known in materials science but was extended to graphs using a Ginzburg--Landau functional and the graph Laplacian. We here generalize the previously proposed model by a non-smooth potential fun…

Estimates smooth graph signals from partial measurements.

New geometric transformations link discrete and continuous curve motions.

Kontsevich's classes distinguish smooth structures on fiber bundles.

Compactness theorems for $G_2$-solitons established with scalar curvature and potential function constraints.

Smooth dec initial data sets may not extend to smooth spacetimes.