Related papers: A note on the hyperbolic singular value decomposit…
For tensor decompositions such as HOSVD and ParaFac, the objective functions are nonconvex. This implies, theoretically, there exists a large number of local optimas: starting from different starting point, the iteratively improved solution…
A hyperbolic system must have a set of linearly independent eigenvectors and corresponding real eigenvalues. In numerical simulations, however, the eigenvalues can be complex because truncation errors pollute a characteristic polynomial of…
Approximating higher-order tensors by the Tucker format has been applied in many fields such as psychometrics, chemometrics, signal processing, pattern classification, and so on. In this paper, we propose some new Tucker-like approximations…
This article focuses on solving the generalized eigenvalue problems (GEP) arising in the source-free Maxwell equation with magnetoelectric coupling effects that models three-dimensional complex media. The goal is to compute the smallest…
We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex…
An efficient, accurate and reliable approximation of a matrix by one of lower rank is a fundamental task in numerical linear algebra and signal processing applications. In this paper, we introduce a new matrix decomposition approach termed…
For any real diagonalizable matrix with complex eigenvalues we provide a real, coordinate free decomposition with a clear geometric interpretation.
This paper provides an advanced mathematical theory of the Generalized Singular Value Decomposition (GSVD) and its applications. We explore the geometry of the GSVD which provides a long sought for ellipse picture which includes a…
In this paper, we address the well-known challenge in the numerical solution of time-fractional partial differential equations (TFPDEs), namely, that the dependence on all previous time levels leads to storage requirements that grow…
Data representation in non-Euclidean spaces has proven effective for capturing hierarchical and complex relationships in real-world datasets. Hyperbolic spaces, in particular, provide efficient embeddings for hierarchical structures. This…
The canonical polyadic decomposition (CPD) is a fundamental tensor decomposition which expresses a tensor as a sum of rank one tensors. In stark contrast to the matrix case, with light assumptions, the CPD of a low rank tensor is…
Tensor decompositions have rich applications in statistics and machine learning, and developing efficient, accurate algorithms for the problem has received much attention recently. Here, we present a new method built on Kruskal's uniqueness…
In this work we demonstrate that SVD-based model reduction techniques known for ordinary differential equations, such as the proper orthogonal decomposition, can be extended to stochastic differential equations in order to reduce the…
We develop an Iterative version of the Singular Value Decomposition (ISVD) that jointly analyzes a finite number of data matrices to identify signals that correlate among the rows of matrices. It will be illustrated how the supervised…
Distributions measured in high energy physics experiments are usually distorted and/or transformed by various detector effects. A regularization method for unfolding these distributions is re-formulated in terms of the Singular Value…
A stationary value based algorithm (SVA) is provided to solve the nearest Kronecker product decomposition (KPD) problem of vector form hypermatrices. Using the algorithm successively, the finite sum KPD is also solved. Then the permutation…
Dynamic Mode Decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of non-linear systems from experimental datasets. Recently, several attempts have extended DMD to the context of low-rank approximations. This…
Orthogonal decomposition of tensors is a generalization of the singular value decomposition of matrices. In this paper, we study the spectral theory of orthogonally decomposable tensors. For such a tensor, we give a description of its…
Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value…
Based on the matrix expression of general nonlinear numerical analogues presented by the present author, this paper proposes a novel philosophy of nonlinear computation and analysis. The nonlinear problems are considered an ill-posed linear…