arXiv research

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

This paper uses the technology of weighted and regular triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Regular triangulations are studied in some detail, including flip algorithms. The Laplacian is then studied as an operator on functions of the vertices as a generalized wei…

Propagation-regularization improves GNN performance by infusing extra graph information.

The p-Laplacian Transformer improves transformer models by assigning higher attention weights to tokens in close proximity.

Dual regularized graph Laplacian improves spectral clustering for community detection.

Method estimates multiple related Gaussian distributions using Laplacian regularization.

This paper tackles the curse of dimensionality in semi-supervised learning using Laplacian regularization.

Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.

The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.

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

Regularization of the classical Laplacian matrices was empirically shown to improve spectral clustering in sparse networks. It was observed that small regularizations are preferable, but this point was left as a heuristic argument. In this paper we formally determine a proper regularization which is intimately related …

A new method for community detection in networks is presented.

Recently, Mahoney and Orecchia demonstrated that popular diffusion-based procedures to compute a quick \emph{approximation} to the first nontrivial eigenvector of a data graph Laplacian \emph{exactly} solve certain regularized Semi-Definite Programs (SDPs). In this paper, we extend that result by providing a statistica…

A new method for few-shot learning using Laplacian regularization.

This paper presents a bias-variance tradeoff of graph Laplacian regularizer, which is widely used in graph signal processing and semi-supervised learning tasks. The scaling law of the optimal regularization parameter is specified in terms of the spectral graph properties and a novel signal-to-noise ratio parameter, whi…

Stratified models depend in an arbitrary way on a selected categorical feature that takes $K$ values, and depend linearly on the other $n$ features. Laplacian regularization with respect to a graph on the feature values can greatly improve the performance of a stratified model, especially in the low-data regime. A sign…

Paper studies minimax optimal regression using Laplacian smoothing over graphs.

Universal inequalities for Laplacian eigenvalues on discrete groups.

Developed a framework for designing filters in spectral GCNNs with improved performance.

The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of Kähler Einstein metrics on Fano manifolds, based on Chen's weak C^{1,1} geodesic without using any further regularities. Unlike the smooth case, the lack of regularities on the geodesic forbids us to use spectral formula of the…

We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci…

Proves tropical Hodge theory for smooth projective varieties, conditional on Laplacian regularity.

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers--Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we …

The paper develops bounds and regularity for minimal boundaries in non-smooth spaces with Ricci curvature.

In this paper, we discuss spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold. First, we give a survey of results on generalized smooth distributions on manifolds, Riemannian structures and associated Laplacians. Then, under the assumption that the singular foliation…

S2MAM improves semi-supervised learning by selecting relevant variables and updating similarity metrics.

In this paper we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold, initiated in a previous paper. Under assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distributi…

Sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R)

In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of…

We propose a new approach, multi-view Laplacian support vector machines (SVMs), for semi-supervised learning under the multi-view scenario. It integrates manifold regularization and multi-view regularization into the usual formulation of SVMs and is a natural extension of SVMs from supervised learning to multi-view sem…

The paper studies the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with flat unitary line bundle.

Study spectral properties of sub-Laplacians in Carnot groups.

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4π$.

This paper investigates the use of methods from partial differential equations and the Calculus of variations to study learning problems that are regularized using graph Laplacians. Graph Laplacians are a powerful, flexible method for capturing local and global geometry in many classes of learning problems, and the tec…

The paper calculates determinants for Laplacians on spinor bundles over surfaces with flat metrics.

We study comparison formulas for $ζ$-regularized determinants of self-adjoint extensions of the Laplacian on flat conical surfaces of genus $g\geq 2$. The cases of trivial and non-trivial holonomy of the metric turn out to differ significantly.

New regularization techniques improve stability of deep neural networks.

BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.

Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.

We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components …

We study the heat trace for both the drifting Laplacian as well as Schrödinger operators on compact Riemannian manifolds. In the case of a finite regularity potential or weight function, we prove the existence of a partial (six term) asymptotic expansion of the heat trace for small times as well as a suitable remainder…

The paper shows equivalent interpretations of Laplacian bounds in RCD spaces.

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues $λ_k$ of conformal sub-Riemannian metrics that are asymptotically sharp as $k\to +\infty$. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for…

It is shown that in a tower of coverings the regularized determinant of a generalized Laplacian converges to the $L^2$-determinant. This shows generic nontriviality of analytic torsion or regularized determinants since the $L^2$-counterparts are easier to compute. We further have an "Euler product expansion" for regula…

This article describes an implementation of a nonparametric Bayesian approach to solving binary classification problems on graphs. We consider a hierarchical Bayesian approach with a prior that is constructed by truncating a series expansion of the soft label function using the graph Laplacian eigenfunctions as basis f…

The paper studies the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with specific boundary conditions.

Study shows exponential growth of Laplacian determinant on random hyperbolic surfaces.

The paper proves boundedness of envelopes in complex manifolds.