arXiv research

Machine learning models solve inverse eigenvalue problems for symmetric potentials and refractive indices.

This is a continuation of our previous work arXiv:1601.05617 on trace and inverse trace of Steklov eigenvalues. More new inequalities for the trace and inverse trace of Steklov eigenvalues are obtained.

Inverse spectral theory reveals shapes from sound.

By studying the monotonicity of the first nonzero eigenvalues of Laplace and p-Laplace operators on a closed convex hypersurface $M^n$ which evolves under inverse mean curvature flow in $\mathbb{R}^{n+1}$, the isoperimetric lower bounds for both eigenvalues were founded.

In this paper, we obtain some new estimates for the trace and inverse trace of Steklov eigenvalues. The estimates generalize some previous results of Hersch-Payne-Schiffer , Brock}, Raulot-Savo and Dittmar.

Paper develops IFTRR to solve sparse generalized eigenvalue problems efficiently.

We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on $M$ is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potenti…

Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.

Stable solution found for manifold topology from boundary data.

The paper studies magnetic field effects on surface eigenvalues and spectral properties.

Recent developments link Steklov eigenvalues to manifold geometry.

Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.

In many contexts the modal properties of a structure change, either due to the impact of a changing environment, fatigue, or due to the presence of structural damage. For example during flight, an aircraft's modal properties are known to change with both altitude and velocity. It is thus important to quantify these cha…

We consider principal component analysis (PCA) in decomposable Gaussian graphical models. We exploit the prior information in these models in order to distribute its computation. For this purpose, we reformulate the problem in the sparse inverse covariance (concentration) domain and solve the global eigenvalue problem …

Study shows stability of Schrödinger operator spectral data on a manifold.

We use methods of random matrix theory to analyze the cross-correlation matrix C of price changes of the largest 1000 US stocks for the 2-year period 1994-95. We find that the statistics of most of the eigenvalues in the spectrum of C agree with the predictions of random matrix theory, but there are deviations for a fe…

The graphical lasso (glasso) is a widely-used fast algorithm for estimating sparse inverse covariance matrices. The glasso solves an L1 penalized maximum likelihood problem and is available as an R library on CRAN. The output from the glasso, a regularized covariance matrix estimate a sparse inverse covariance matrix e…

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

Study elliptic operators on glued manifolds, reducing to finite-dimensional systems.

Bayesian approach learns linear operators from noisy data.

New rigidity result for Steklov eigenvalues on manifolds.

We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special structure superimposed due to the time-shift. We demonstrate that the associated eigen…

Extends dimension reduction to data-driven settings without gradients.

In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $Σ$ into two components and we know the eigenvalues $λ_j$ of the Laplace ope…

The classical inverse problem of recovering a simply connected smooth planar domain from the Steklov spectrum \cite{E} is equivalent to the problem of recovering, up to a conformal equivalence, a positive function $a\in C^\infty({\mathbb S})$ on the unit circle ${\mathbb S}=\{e^{iθ}\}$ from the eigenvalue spectrum of t…

A new data-adaptive prior stabilizes kernel learning in operators.

One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs). We address this problem by taking advantage of recent advances in scientific machine learning and the dynamically orthogonal (DO) and bi-orthogonal (BO) methods for representing …

Researchers approximate spectral targets on manifolds with constant negative curvature.

Exact risk and learning rate curves derived for adaptive SGD on high-dimensional problems.

The paper defines surface area for graphs and derives spectral estimates.

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlat…

Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise. The maximum likelihood solution for the model is an eigenvalue problem on the sample covariance matrix. In this paper we consider the situation where the data v…

The L1-regularized maximum likelihood estimation problem has recently become a topic of great interest within the machine learning, statistics, and optimization communities as a method for producing sparse inverse covariance estimators. In this paper, a proximal gradient method (G-ISTA) for performing L1-regularized co…

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also prov…

Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $Δ_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We t…

Penrose conjecture proven for specific initial data sets.

Researchers compute the spectrum of Hodge-Laplacian on 1-forms for SU(2) and SO(3).

Grosjean proved that the $(1/p)$-th power of the first eigenvalue of the $p$-Laplacian on a closed Riemannian manifold converges to the twice of the inverse of the diameter of the space, as $p \to \infty$. Before this, a corresponding result for the Dirichlet first eigenvalues was also obtained by Juutinen, Lindqvist a…

We introduce a distributionally robust maximum likelihood estimation model with a Wasserstein ambiguity set to infer the inverse covariance matrix of a $p$-dimensional Gaussian random vector from $n$ independent samples. The proposed model minimizes the worst case (maximum) of Stein's loss across all normal reference d…

Study non-asymptotic bounds on correlation in high-dimensional linear systems, revealing invariant subspaces and bottlenecks.

The paper sets lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted La…

We study the basic problem of robust subspace recovery. That is, we assume a data set that some of its points are sampled around a fixed subspace and the rest of them are spread in the whole ambient space, and we aim to recover the fixed underlying subspace. We first estimate "robust inverse sample covariance" by solvi…

We consider large scale empirical risk minimization (ERM) problems, where both the problem dimension and variable size is large. In these cases, most second order methods are infeasible due to the high cost in both computing the Hessian over all samples and computing its inverse in high dimensions. In this paper, we pr…

The paper solves the Steklov spectral inverse problem for conformal metrics.

New biharmonic Steklov problem on forms yields eigenvalue estimates.

The paper provides estimates for eigenvalues of elliptic differential problems.

We analyzed cross-correlations between price fluctuations of global financial indices (20 daily stock indices over the world) and local indices (daily indices of 200 companies in the Korean stock market) by using random matrix theory (RMT). We compared eigenvalues and components of the largest and the second largest ei…