arXiv research

The paper proves inequalities for Steklov eigenvalues on finite graphs.

Variational method for eigenvalues on manifolds.

Recall that Federer-Fleming defined the notion of flat convergence of submanifolds of Euclidean space to solve the Plateau problem. Here we prove the upper semicontinuity of Neumann eigenvalues of the submanifolds when they converge in the flat sense without losing volume. With an additional condition on the boundaries…

Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.

Graph Laplacian approximates manifold eigenvalues with controlled curvature bounds.

The extragradient method accelerates convergence in complex game dynamics.

Eigenvalues of random hyperbolic surface covers converge to hyperbolic plane's.

The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.

Study approximates product of spheres using Laplacian eigenvalues.

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eige…

Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.

Spheres' spectral structure converges to Gaussian space's as dimensions grow.

Active data collection improves convergence rates in operator learning.

The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.

The paper studies eigenvalue problems on manifolds and recovers known inequalities.

We relate small 1-form Laplacian eigenvalues to relative cycle complexity on closed hyperbolic manifolds: small eigenvalues correspond to closed geodesics no multiple of which bounds a surface of small genus. We describe potential applications of this equivalence principle toward proving optimal torsion homology growth…

We study the limiting behavior of eigenfunctions/eigenvalues of the Laplacian of a family of Riemannian metrics that degenerates on a hypersurface. Our results generalize earlier work concerning the degeneration of hyperbolic surfaces.

Study on eigenvalue distribution of correlated time series, showing deformation of Marchenko-Pastur distribution.

Constructs metrics with zero eigenvalues for Hodge-Laplacian.

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a…

Gradient descent proves global convergence for 4-layer matrix factorization.

The paper studies free boundary minimal surfaces with many boundaries and their convergence to closed minimal surfaces.

By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the …

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. Consider the points $X_1, X_2,..., X_n$ are vectors drawn i.i.d. from a distribution with mean zero and covariance $Σ$, where $Σ$ is unknown. Let $A_n = X_nX_n^T$, then $E[A_n] = Σ$. This paper …

We derive various pinching results for small Dirac eigenvalues using the classification of $\text{spin}^c$ and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for $\text{spin}^c$ manifolds which involves a general study on convergence of Riemannian manifolds with a pr…

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…

Study on geodesics and eigenvalues on random hyperbolic surfaces with cusps.

Rough and Hodge Laplacians eigenvalues approach zero with fixed volume.

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compar…

Study on eigenvalue distribution of correlated time series deforming the semi-circle law.

Improved convergence speed of principal component analysis through modified learning rules.

Study eigenvalues of Laplace operator on specific 3D manifolds under Ricci flow.

In this paper, we define a new capacity which allows us to control the behaviour of the Dirichlet spectrum of a compact Riemannian manifold with boundary, with "small" subsets (which may intersect the boundary) removed. This result generalizes a classical result of Rauch and Taylor ("the crushed ice theorem"). In the s…

Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on …

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of…

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. …

We propose a method to clean covariance matrices of nonstationary systems by using time-independent eigenvalues.

SGD benefits from a directional bias in kernel regression models.

We show the convergence properties of the eigenvalues of the Dirac operator on a spin manifold with a Riemannian flow when the metric is collapsed along the flow.

Study eigenvalues and eigenvectors in neural networks, focusing on signal propagation.

In this paper, we consider the sparse eigenvalue problem wherein the goal is to obtain a sparse solution to the generalized eigenvalue problem. We achieve this by constraining the cardinality of the solution to the generalized eigenvalue problem and obtain sparse principal component analysis (PCA), sparse canonical cor…

New bounds on KPCA efficiency reveal conditions for fast convergence.

We study the behavior of the spectrum of the Dirac operator on collapsing S^1-bundles. Convergent eigenvalues will exist if and only if the spin structure is projectable.

Study the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.

The paper finds large Steklov eigenvalues on manifolds using homogenization.

We discuss the behavior of $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$. Applications include new estimates for the first eigenvalues of the $p$-Laplacian on Rieman…

In this paper, we extend the Witten-Helffer-Sjöstrand theory from Morse functions to generalized Morse functions. In this case, the spectrum of the Witten deformed Laplacian $Δ(t)$, for large t, can be seperated into the small eigenvalues (which tend to 0 as $t\rightarrow\infty$), large and very large eigenvalues (both…

Let $(M,J)$ be a compact complex manifold of complex dimension $m$ and let $g_s$ be a one-parameter family of Hermitian forms on $M$ that are smooth and positive definite for each fixed $s\in (0,1]$ and that somehow degenerates to a Hermitian pseudometric $h$ for $s$ tending to $0$. In this paper under rather general a…