Related papers: Computing generalized inverses using LU factorizat…
Although the vectorization operation is known and well-defined, it is only defined for 2-D matrices, and its inverse isn't as well-popularized. This work proposes to generalize the vectorization to higher dimensions, and define…
In this paper we provide an algorithm, similar to the simplex algorithm, which determines a rational cp-factorization of a given matrix, whenever the matrix allows such a factorization. This algorithm can be used to show that every integral…
Reverse order law for the Moore-Penrose inverses of tensors are useful in the field of multilinear algebra. In this paper, we first prove some more identities involving the Moore-Penrose inverse of tensors. We then obtain a few necessary…
This is the second part of a two-paper series on generalized inverses that minimize matrix norms. In Part II we focus on generalized inverses that are minimizers of entrywise p norms whose main representative is the sparse pseudoinverse for…
Many applications in scientific computing and data science require the computation of a rank-revealing factorization of a large matrix. In many of these instances the classical algorithms for computing the singular value decomposition are…
The computational complexity of simultaneous inference methods in high-dimensional linear regression models quickly increases with the number variables. This paper proposes a computationally efficient method based on the Moore-Penrose…
This paper presents a sequential randomized lowrank matrix factorization approach for incrementally predicting values of an unknown function at test points using the Gaussian Processes framework. It is well-known that in the Gaussian…
We describe an efficient parallel implementation of the selected inversion algorithm for distributed memory computer systems, which we call \texttt{PSelInv}. The \texttt{PSelInv} method computes selected elements of a general sparse matrix…
Here we discuss a regularized version of the factorization method for positive operators acting on a Hilbert Space. The factorization method is a qualitative reconstruction method that has been used to solve many inverse shape problems. In…
We examine a special case of the multilevel factor model, with covariance given by multilevel low rank (MLR) matrix~\cite{parshakova2023factor}. We develop a novel, fast implementation of the expectation-maximization algorithm, tailored for…
The notion of a Moore-Penrose inverse (M-P inverse) was introduced by Moore in 1920 and rediscovered by Penrose in 1955. The M-P inverse of a complex matrix is a special type of inverse which is unique, always exists, and can be computed…
In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great…
If a tensor with various symmetries is properly unfolded, then the resulting matrix inherits those symmetries. As tensor computations become increasingly important it is imperative that we develop efficient structure preserving methods for…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
This manuscript proposes a generalized inverse for a dual matrix called dual Drazin generalized inverse (DDGI) which generalizes the notion of the dual group generalized inverse (DGGI). Under certain necessary and sufficient conditions, we…
Matrices with off-diagonal decay appear in a variety of fields in mathematics and in numerous applications, such as signal processing, statistics, communications engineering, condensed matter physics, and quantum chemistry. Numerical…
Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. Unlike binary matrix factorization based on standard arithmetic, BMF employs the Boolean OR and AND operations for the…
The algebraic theory of third-order tensors under the $t$-product is naturally formulated over the complex field via Fourier block diagonalization. However, many applications require real-valued representations. In this paper, we…
In this paper we present a method for matrix inversion based on Cholesky decomposition with reduced number of operations by avoiding computation of intermediate results; further, we use fixed point simulations to compare the numerical…
In this paper, we revisit implicit regularization from the ground up using notions from dynamical systems and invariant subspaces of Morse functions. The key contributions are a new criterion for implicit regularization---a leading…