Related papers: Perspectives on CUR Decompositions
A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties…
This paper is concentrated on the classification of permutation matrix with the permutation similarity relation, mainly about the canonical form of a permutational similar equivalence class, the cycle matrix decomposition of a permutation…
We characterize some variations of pseudoskeleton (also called CUR) decompositions for matrices and tensors over arbitrary fields. These characterizations extend previous results to arbitrary fields and to decompositions which use…
We consider the factorization of a rectangular matrix $X $ into a positive linear combination of rank-one factors of the form $u v^\top$, where $u$ and $v$ belongs to certain sets $\mathcal{U}$ and $\mathcal{V}$, that may encode specific…
Factor analysis, often regarded as a Bayesian variant of matrix factorization, offers superior capabilities in capturing uncertainty, modeling complex dependencies, and ensuring robustness. As the deep learning era arrives, factor analysis…
The CUR decomposition of an $m \times n$ matrix $A$ finds an $m \times c$ matrix $C$ with a subset of $c < n$ columns of $A,$ together with an $r \times n$ matrix $R$ with a subset of $r < m$ rows of $A,$ as well as a $c \times r$ low-rank…
Based on the column pivoted QR decomposition, we propose some randomized algorithms including pass-efficient ones for the generalized CUR decompositions of matrix pair and matrix triplet. Detailed error analyses of these algorithms are…
This paper derives the CUR-type factorization for tensors in the Tucker format based on a new variant of the discrete empirical interpolation method known as L-DEIM. This novel sampling technique allows us to construct an efficient…
The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem.…
In this note we mainly study the fine Jordan-Chevalley decomposition: a refinement of the classical Jordan-Chevalley decomposition of a matrix and we pay a particular attention to the field of the coefficients of the matrix. Moreover we…
This text investigates relations between two well-known family of algorithms, matrix factorisations and recursive linear filters, by describing a probabilistic model in which approximate inference corresponds to a matrix factorisation…
The singular value decomposition (SVD) is commonly used in applications requiring a low rank matrix approximation. However, the singular vectors cannot be interpreted in terms of the original data. For applications requiring this type of…
We study the number of factorizations of a positive integer, where the parts of the factorization are of l different colors (or kinds). Recursive or explicit formulas are derived for the case of unordered and ordered, distinct and…
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 study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
We present a new restricted SVD-based CUR (RSVD-CUR) factorization for matrix triplets $(A, B, G)$ that aims to extract meaningful information by providing a low-rank approximation of the three matrices using a subset of their rows and…
We introduce the $D$-decomposition, a non-orthogonal matrix factorization of the form $A \approx P D Q$, where $P \in \mathbb{R}^{n \times k}$, $D \in \mathbb{R}^{k \times k}$, and $Q \in \mathbb{R}^{k \times n}$. The decomposition is…
This work investigates the accuracy and numerical stability of CUR decompositions with oversampling. The CUR decomposition approximates a matrix using a subset of columns and rows of the matrix. When the number of columns and the rows are…
Dimension reduction is an essential tool for analyzing high dimensional data. Most existing methods, including principal component analysis (PCA), as well as their extensions, provide principal components that are often linear combinations…
This article studies how to form CUR decompositions of low-rank matrices via primarily random sampling, though deterministic methods due to previous works are illustrated as well. The primary problem is to determine when a column submatrix…