arXiv research

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

The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.

The famous Whitney formula relates the winding number of the smooth generic curve in the real plane to the number of its self-intersection points counted with appropriate signs. We extend this formula to smooth immersions of R^n to R^{2n}. Then use this result together with the general technique of Laplace integrals to…

New upper bound found for nodal sets of Laplace eigenfunctions.

Study finds eigenvalue patterns on rough manifolds with measurable metrics.

The paper studies graph Laplace operator behavior near isolated singularities.

Deep neural kernels and Laplace kernel have equivalent RKHS on spheres.

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank three Lie groups with biinvariant Riemannian metric and established a connection of obtained formulas with the number theory and integer ternary and binary quadratic f…

Study on $p$-Laplace equation in convex cones, proving rigidity under specific conditions.

Neural Laplace models diverse DEs in the Laplace domain for better dynamics.

Anisotropic metric on manifolds uniquely determined by boundary data.

Private two-sample tests under LDP achieve minimax rates for multinomial and continuous data.

Review of Yau's conjecture on zero sets of Laplace eigenfunctions.

A new algorithm selects independent coordinates for complex manifolds.

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank four Lie groups with biinvariant Riemannian metric, corresponding to root systems $B_4$, $C_4$, $D_4$ and established a connection of obtained formulas with the number…

We show that any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Lapla…

The paper proves a new method to find the maximum Laplace eigenvalues on surfaces.

Study of loops in sums of Laplace eigenfunctions on surfaces.

This paper computes a heat coefficient for nonminimal Laplace type operators.

Laplace kernel and Neural Tangent Kernels are shown to be nearly identical for normalized data.

In this thesis we study the geometry of the fixed point set $Σ$ of a smooth mapping $Φ: M\to M$ on a smooth compact Riemannian manifold $M$ without boundary by computing the asymptotic expansion of the deformed heat trace $\Trace Φ\exp(tΔ)$ of the Laplace operator $Δ$ on $M$. We assume that the fixed point set $Σ$ is a…

Logistic Gaussian process (LGP) priors provide a flexible alternative for modelling unknown densities. The smoothness properties of the density estimates can be controlled through the prior covariance structure of the LGP, but the challenge is the analytically intractable inference. In this paper, we present approximat…

Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $\varphi_λ: Δ_M \varphi_λ+ λ\varphi_λ= 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $λ$. We show that $H^{n-1}(\{ \varphi_λ=0\}) \leq C λ^α,$ where $α>1/2$ is a constant, whi…

Estimates the index of the Laplace operator on planar domains with Robin boundary condition.

Several numerical approximation strategies for the expectation-propagation algorithm are studied in the context of large-scale learning: the Laplace method, a faster variant of it, Gaussian quadrature, and a deterministic version of variational sampling (i.e., combining quadrature with variational approximation). Exper…

We make a computational study to know what kind of isospectralities among lens spaces and lens orbifolds exist considering the Hodge--Laplace operators acting on smooth $p$-forms. Several evidenced facts are proved and some others are conjectured.

Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $H^{n-1}(\{u=0 \}\cap B) \geq c$. We prove Nadirashvili's conjecture as well as its counterpar…

The paper proves eigenvalues are simple for specific operators on bundles.

Study new spectral invariants for Laplace and Dirac operators on manifolds.

Proposes an algorithm for semi-supervised learning with budget constraints.

Formula for Laplacian determinants on polygonal domains with slits.

Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a hyperbolic metric $σ$ of total area $A$. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area …

Study on lengths and curvatures of harmonic functions on smooth and singular surfaces.

Bayesian framework for sequential learning tasks with low-rank approximations.

New heat dispersion laws established for smooth compact manifolds.

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator …

This purpose of this write-up is to share an idea for accurate computation of Laplace eigenvalues on a broad class of smooth domains. We represent the eigenfunction $u$ as a linear combination of eigenfunctions corresponding to the common eigenvalue $ρ^{2}$:\EQN{6}{1}{}{0}{\RD{\CELL{u(r,θ) =\sum_{n=0}^{N}P_{n}J_{n}(ρ) …

The study optimizes distribution estimation from samples with relative entropy error, adapting to sparse distributions.

GES algorithm improves consistency for nonparametric DAG models.

The Alesker-Poincare pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pai…

Generalizes results for Riemannian manifolds with boundary to those without.

The paper bounds Cheeger ratios of eigenfunctions and their level sets.

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some coni…

Eigenvalues on spheres are compared to the unit round sphere, proving a sharp bound and equality condition.

The paper shows Gaussian fluctuations in eigenvalue statistics of random hyperbolic surfaces.

Paper studies Frobenius manifolds and quantum differential equations, proving Dubrovin Conjecture for Hirzebruch surfaces.

We propose a Laplace approximation that creates a stochastic unit from any smooth monotonic activation function, using only Gaussian noise. This paper investigates the application of this stochastic approximation in training a family of Restricted Boltzmann Machines (RBM) that are closely linked to Bregman divergences.…

Introduces new Hermitian metrics linking to Gauduchon and balanced metrics.