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Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is…

The tensor t-product, introduced by Kilmer and Martin [26], is a powerful tool for the analysis of and computation with third-order tensors. This paper introduces eigentubes and eigenslices of third-order tensors under the t-product. The…

Eigensolvers involving complex moments can determine all the eigenvalues in a given region in the complex plane and the corresponding eigenvectors of a regular linear matrix pencil. The complex moment acts as a filter for extracting…

A fixed-time stable dynamical system for solving the extended vertical linear complementarity problem (EVLCP) is developed. The system is based on the reformulation of EVLCP as a special case of a new kind of generalized absolute value…

The generalized eigenvalue (GE) problems are of particular importance in various areas of science engineering and machine learning. We present a variational quantum algorithm for finding the desired generalized eigenvalue of the GE problem,…

This paper is concerned with the computable error estimates for the eigenvalue problem which is solved by the general conforming finite element methods on the general meshes. Based on the computable error estimate, we can give an…

We consider the problem of low canonical polyadic (CP) rank tensor completion. A completion is a tensor whose entries agree with the observed entries and its rank matches the given CP rank. We analyze the manifold structure corresponding to…

Recently, many structured tensors are defined and their properties are discussed in the literature. In this paper, we introduce a new class of structured tensors, called exceptionally regular tensor, which is relevant to the tensor…

Tensor decomposition is a fundamental unsupervised machine learning method in data science, with applications including network analysis and sensor data processing. This work develops a generalized canonical polyadic (GCP) low-rank tensor…

In this paper, we compute the H- and Z-eigenvalues of even order symmetric tensors by using the adaptive cubic regularization algorithm.

In this article we combine the projective Landweber method, recently proposed by the authors, with Kaczmarz's method for solving systems of non-linear ill-posed equations. The underlying assumption used in this work is the tangential cone…

We propose a novel rank-adaptive higher-order orthogonal iteration (HOOI) algorithm to compute the truncated Tucker decomposition of higher-order tensors with a given error tolerance, and prove that the method is locally optimal and…

An algorithm for finding the eigenvalue of a nonnegative irreducible tensor was recently proposed by Michael Ng, Liqun Qi, and Guanglu Zhou in {\it Finding the largest eigenvalue of a nonnegative tensor}. However, the authors did not prove…

Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To…

We develop a systematic way to solve linear equations involving tensors of arbitrary rank. We start off with the case of a rank $3$ tensor, which appears in many applications, and after finding the condition for a unique solution we derive…

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor…

We propose a tensor generalized approximate message passing (TeG-AMP) algorithm for low-rank tensor inference, which can be used to solve tensor completion and decomposition problems. We derive TeG-AMP algorithm as an approximation of the…

Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical…

Tensor completion is a technique of filling missing elements of the incomplete data tensors. It being actively studied based on the convex optimization scheme such as nuclear-norm minimization. When given data tensors include some noises,…

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…