Related papers: Elimination and Factorization
Let $H$ be a numerical semigroup minimally generated by an almost arithmetic sequence. We give a description of a possible row-factorization $(\RF)$ matrix for each pseudo-Frobenius element of $H.$ Further, when $H$ is symmetric and has…
Sparse matrix reordering can significantly reduce the fill-in during matrix factorization, thereby decreasing the computational and storage requirements in sparse matrix computations. Finding a minimal fill-in ordering is known to be an…
The ability to represent complex high dimensional probability distributions in a compact form is one of the key insights in the field of graphical models. Factored representations are ubiquitous in machine learning and lead to major…
We discuss an interesting duality known to occur for certain complex reflection groups, namely the duality groups. Our main construction yields a concrete, representation theoretic realisation of this duality. This allows us to naturally…
Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…
We present a new proof of the well known formula for the rank of the inclusion matrix by constructing a $k\mathcal{S}_n$-module spanned by the columns of this matrix and calculating its dimension.
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…
Nonnegative matrix factorization (NMF) has become a widely used tool for the analysis of high-dimensional data as it automatically extracts sparse and meaningful features from a set of nonnegative data vectors. We first illustrate this…
Given labeled data represented by a binary matrix, we consider the task to derive a Boolean matrix factorization which identifies commonalities and specifications among the classes. While existing works focus on rank-one factorizations…
The Leinster matrix corresponding to a finite category has entries counting the number of morphisms between objects. A first question is to know which positive integer matrices come from at least one finite category. Here, that question…
Predicting unobserved entries of a partially observed matrix has found wide applicability in several areas, such as recommender systems, computational biology, and computer vision. Many scalable methods with rigorous theoretical guarantees…
The row (column) completion problem of polynomial matrices of given degree with prescribed eigenstructure has been studied in \cite{AmBaMaRo23}, where several results of prescription of some of the four types of invariants that form the…
Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. It aims to find a low rank approximation for nonnegative data M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to…
Let $\mathbf{A}_{n,m;k}$ be a random $n \times m$ matrix with entries from some field $\mathbb{F}$ where there are exactly $k$ non-zero entries in each column, whose locations are chosen independently and uniformly at random from the set of…
This paper examines the problem of locating outlier columns in a large, otherwise low-rank, matrix. We propose a simple two-step adaptive sensing and inference approach and establish theoretical guarantees for its performance; our results…
In this letter, we propose a new identification criterion that guarantees the recovery of the low-rank latent factors in the nonnegative matrix factorization (NMF) model, under mild conditions. Specifically, using the proposed criterion, it…
The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm…
Given a graph or similarity matrix, we consider the problem of recovering a notion of true distance between the nodes, and so their true positions. We show that this can be accomplished in two steps: matrix factorisation, followed by…
We propose a generalization of the factorization method to the case when $\mathcal{G}$ is a finite dimensional Lie algebra such that $\mathcal{G}=\mathcal{G}_0\oplus M \oplus N$ (direct sum of vector spaces), where $\mathcal{G}_0$ is a…
The Kronecker product is an invaluable tool for data-sparse representations of large networks and matrices with countless applications in machine learning, graph theory and numerical linear algebra. In some instances, the sparsity pattern…