Related papers: Spectral Methods for Matrices and Tensors

Spectral tetris is a fexible and elementary method to construct unit norm frames with a given frame operator, having all of its eigenvalues greater than or equal to two. One important application of spectral tetris is the construction of…

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on…

We propose a sampling-based method for computing the tensor ring (TR) decomposition of a data tensor. The method uses leverage score sampled alternating least squares to fit the TR cores in an iterative fashion. By taking advantage of the…

We present an overarching framework for stable spectral methods on a triangle, defined by a multivariate W-system and based on orthogonal polynomials on the triangle. Motivated by the Koornwinder orthogonal polynomials on the triangle, we…

Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or…

We present a spectral analysis for matrix scaling and operator scaling. We prove that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent…

A spectral method is considered for approximating the fractional Laplacian and solving the fractional Poisson problem in 2D and 3D unit balls. The method is based on the explicit formulation of the eigenfunctions and eigenvalues of the…

We give an efficient algorithm which can obtain a relative error approximation to the spectral norm of a matrix, combining the power iteration method with some techniques from matrix reconstruction which use random sampling.

In an era of unprecedented deluge of (mostly unstructured) data, graphs are proving more and more useful, across the sciences, as a flexible abstraction to capture complex relationships between complex objects. One of the main challenges…

We investigate how to solve smooth matrix optimization problems with general linear inequality constraints on the eigenvalues of a symmetric matrix. We present solution methods to obtain exact global minima for linear objective functions,…

X-ray spectroscopy is a powerful technique for the analysis of the energy distribution of X-rays from astrophysical sources. It allows for the study of the properties, composition, and physical processes taking place at the site of…

Tensor networks provide an efficient approximation of operations involving high dimensional tensors and have been extensively used in modelling quantum many-body systems. More recently, supervised learning has been attempted with tensor…

The subdifferential of convex functions of the singular spectrum of real matrices has been widely studied in matrix analysis, optimization and automatic control theory. Convex analysis and optimization over spaces of tensors is now gaining…

This paper discusses a general method for spectral type theorems using metric spaces instead of vector spaces. Advantages of this approach are that it applies to genuinely non-linear situations and also to random versions. Metric analogs of…

Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets…

Isospectral reduction is an important tool for network/matrix analysis as it reduces the dimension of a matrix/network while preserving its eigenvalues and eigenvectors. The main contribution of this manuscript is a proposed algorithmic…

Computing eigenvalues of very large matrices is a critical task in many machine learning applications, including the evaluation of log-determinants, the trace of matrix functions, and other important metrics. As datasets continue to grow in…

Spectral residual methods are derivative-free and low-cost per iteration procedures for solving nonlinear systems of equations. They are generally coupled with a nonmonotone linesearch strategy and compare well with Newton-based methods for…

We propose a second-order accurate method to estimate the eigenvectors of extremely large matrices thereby addressing a problem of relevance to statisticians working in the analysis of very large datasets. More specifically, we show that…

We consider the problem of parameter estimation in a high-dimensional generalized linear model. Spectral methods obtained via the principal eigenvector of a suitable data-dependent matrix provide a simple yet surprisingly effective…