Related papers: Matrix Decomposition and Applications
In 1954, Alston S. Householder published \textit{Principles of Numerical Analysis}, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower…
Multilinear and tensor decompositions are a popular tool in linear and multilinear algebra and have a wide range of important applications to modern computing. Our paper of 1972 presented the first nontrivial application of such…
Matrix decompositions are fundamental tools in the area of applied mathematics, statistical computing, and machine learning. In particular, low-rank matrix decompositions are vital, and widely used for data analysis, dimensionality…
We take matrix decompositions that are usually applied to matrices over the real numbers or complex numbers, and extend them to matrices over an algebra called the double numbers. In doing so, we unify some matrix decompositions: For…
The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important…
Over the past decades, the increasing dimensionality of data has increased the need for effective data decomposition methods. Existing approaches, however, often rely on linear models or lack sufficient interpretability or flexibility. To…
Matrix factorization is an important mathematical problem encountered in the context of dictionary learning, recommendation systems and machine learning. We introduce a new `decimation' scheme that maps it to neural network models of…
We present a motivating example for matrix multiplication based on factoring a data matrix. Traditionally, matrix multiplication is motivated by applications in physics: composing rigid transformations, scaling, sheering, etc. We present an…
Matrix multiplication (hereafter we use the acronym MM) is among the most fundamental operations of modern computations. The efficiency of its performance depends on various factors, in particular vectorization, data movement and arithmetic…
We prove the decomposition of arbitrary diagonal operators into tensor and matrix products of smaller matrices, focusing on the analytic structure of the resulting formulas and their inherent symmetries. Diagrammatic representations are…
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which…
Low-rank approximations are essential in modern data science. The interpolative decomposition provides one such approximation. Its distinguishing feature is that it reuses columns from the original matrix. This enables it to preserve matrix…
Deep neural network architectures have recently produced excellent results in a variety of areas in artificial intelligence and visual recognition, well surpassing traditional shallow architectures trained using hand-designed features. The…
Low-rank decomposition has emerged as a vital tool for enhancing parameter efficiency in neural network architectures, gaining traction across diverse applications in machine learning. These techniques significantly lower the number of…
We present a natural generalization of the recent low rank + sparse matrix decomposition and consider the decomposition of matrices into components of multiple scales. Such decomposition is well motivated in practice as data matrices often…
Diagonalization, or eigenvalue decomposition, is very useful in many areas of applied mathematics, including signal processing and quantum physics. Matrix decomposition is also a useful tool for approximating matrices as the product of a…
LU-factorization of matrices is one of the fundamental algorithms of linear algebra. The widespread use of supercomputers with distributed memory requires a review of traditional algorithms, which were based on the common memory of a…
Tensors are multidimensional arrays of numerical values and therefore generalize matrices to multiple dimensions. While tensors first emerged in the psychometrics community in the $20^{\text{th}}$ century, they have since then spread to…
A decomposition principle for nonlinear dynamic compartmental systems is introduced in the present paper. This theory is based on the mutually exclusive and exhaustive, analytical and dynamic, novel system and subsystem partitioning…
A large number of computational and scientific methods commonly require decomposing a sparse matrix into triangular factors as LU decomposition. A common problem faced during this decomposition is that even though the given matrix may be…