arXiv research

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…

Proposes a faster Isomap algorithm by reducing eigenvalue decomposition complexity.

Paper describes eigenvalues of genus 3 surfaces graphs.

Proposes a Gaussian process for Koopman mode decomposition.

We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on Kähler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on Kähler manifolds with positive Ricci curvature.

Paper proposes a new optimization framework for learning eigenfunctions of operators.

The paper derives upper bounds on eigenvalues of Laplace-Beltrami operator on hyperbolic surfaces.

Paper identifies key function spaces for ReLU networks based on Fisher information.

Geometric bounds for low Steklov eigenvalues on hyperbolic surfaces with boundaries.

A novel hypergraph partitioning method using tensor eigenvalue decomposition captures super-dyadic interactions.

Paper uses autoencoders for efficient reduced-order modeling of eigenvalue problems.

We discuss the decomposition of the zeta-determinant of the square of the Dirac operator into contributions coming from the different parts of the manifold. The easy case was worked in the previous paper of authors. Due to the assumptions made on the operators in the previous paper, we were able to avoid the presence o…

The paper bounds eigenvalues and integrals of eigenfunctions on hyperbolic manifolds.

Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributi…

Let $M$ be a finite volume oriented Riemannian manifold of dimension $n\geq 3$ and curvature in $[-b^2,-1]$, with thick-thin decomposition $M=M(thick)\cup M(thin)$. Denote by $λ_k(M(thick))$ the k-th eigenvalue for the Laplacian on $M(thick)$, with Neumann boundary conditdions. We show that $λ_k(M(thick))/3\leq λ_k(M)$…

We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We…

In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Trace and inverse trace of Steklov eigenvalues. J. Differential Equations 261 (2016), no. 3, 2026--2040.). Namely, we show that equality of the estimate holds if and only if the manifold …

Eigen-decomposition simplifies quadratic programming with equality constraints.

Revisits orbital minimization for neural operator decomposition.

Two methods are proposed to filter correlations in DCC-GARCH residuals for foreign exchange rates.

Graph Neural Networks outperform the Weisfeiler-Lehman algorithm in representation power.

We propose a method to learn causal response representations through direct effect analysis.

In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenböck decomposition and we study it using the analytical me…

Study examines how risk tolerance impacts long-term investment returns.

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly perturbed version $\mathbf{M}$ is observed. The noise matrix $\mathbf{M}-\mathbf{M}^{\s…

The expectation-maximization (EM) algorithm is an iterative method for finding maximum likelihood estimates when data are incomplete or are treated as being incomplete. The EM algorithm and its variants are commonly used for parameter estimation in applications of mixture models for clustering and classification. This …

A fair PCA method using JEVD ensures balanced data representation.

In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comp…

A new metric compares dynamical systems using operator eigenvalues.

Enhances Koopman operator estimation with intrinsic observables in RKHS.

Complex frequency generalizes eigenvalues in LTI systems.

The study examines Fisher information matrices and neural tangent kernels for simple ReLU networks with random weights.

New method for calculating HOMFLY polynomials in symmetric representations.

The paper extends a spectral evolution model for link prediction in evolving networks.

We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\leq p \leq n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\leq p<n/2$.

Researchers decompose curvature to confirm Hopf conjecture and prove new rigidity theorems.

For a closed Riemannian orbifold $O$, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain $U$ in $O$ whose boundary is a smooth manifold. Generalizing results of several authors, we prove that the metric of $O$ can be…

Derives adjoint formulas for matrix operations and applies them to specific cases.

Extreme classification problems are multiclass and multilabel classification problems where the number of outputs is so large that straightforward strategies are neither statistically nor computationally viable. One strategy for dealing with the computational burden is via a tree decomposition of the output space. Whil…

Study of a $G_2$-equivariant octonionic operator and its right spectrum.

Proposes neural dynamic mode decomposition for end-to-end modeling of nonlinear dynamics.

Extends matrix factorization for deviance-based losses with GLM theory.

A $k$-reflection of the $n$-dimensional complex hyperbolic space ${\rm H}_{\C}^n$ is an element in ${\rm U}(n,1)$ with negative type eigenvalue $λ$, $|λ|=1$, of multiplicity $k+1$ and positive type eigenvalue $1$ of multiplicity $n-k$. We prove that a holomorphic isometry of ${\rm H}_{\C}^n$ is a product of at most fou…

The paper finds shape modes for vortices in a specific sigma model.

Stochastic discount factor (SDF) processes in dynamic economies admit a permanent-transitory decomposition in which the permanent component characterizes pricing over long investment horizons. This paper introduces an empirical framework to analyze the permanent-transitory decomposition of SDF processes. Specifically, …

The paper explores how polynomial roots and operator eigenvalues change with parameters.

Efficiently infers sparse networks from count data with reduced memory usage.

Joint blind source separation (J-BSS) is an emerging data-driven technique for multi-set data-fusion. In this paper, J-BSS is addressed from a tensorial perspective. We show how, by using second-order multi-set statistics in J-BSS, a specific double coupled canonical polyadic decomposition (DC-CPD) problem can be formu…