Related papers: LDU factorization
The success of matrix factorizations such as the singular value decomposition (SVD) has motivated the search for even more factorizations. We catalog 53 matrix factorizations, most of which we believe to be new. Our systematic approach,…
In addition to recent developments in computing speed and memory, methodological advances have contributed to significant gains in the performance of stochastic simulation. In this paper, we focus on variance reduction for matrix…
In this paper we describe a deep learning--based probabilistic algorithm for integer factorisation. We use Lawrence's extension of Fermat's factorisation algorithm to reduce the integer factorisation problem to a binary classification…
We propose a unified framework to solve general low-rank plus sparse matrix recovery problems based on matrix factorization, which covers a broad family of objective functions satisfying the restricted strong convexity and smoothness…
Gaussian elimination with full pivoting generates a PLUQ matrix decomposition. Depending on the strategy used in the search for pivots, the permutation matrices can reveal some information about the row or the column rank profiles of the…
Matrix Factorization (MF) has been widely applied in machine learning and data mining. A large number of algorithms have been studied to factorize matrices. Among them, stochastic gradient descent (SGD) is a commonly used method.…
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…
ReLU matrix decomposition (RMD) is the following problem: given a sparse, nonnegative matrix $X$ and a factorization rank $r$, identify a rank-$r$ matrix $\Theta$ such that $X\approx \max(0,\Theta)$. RMD is a particular instance of…
A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this…
Non-negative matrix factorization with the generalized Kullback-Leibler divergence (NMF) and latent Dirichlet allocation (LDA) are two popular approaches for dimensionality reduction of non-negative data. Here, we show that NMF with…
Polynomial factorization and root finding are among the most standard themes of computational mathematics. Yet still, little has been done for polynomials over quaternion algebras, with the single exception of Hamiltonian quaternions for…
Interested in formalizing the generation of fast running code for linear algebra applications, the authors show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category…
Discrete transforms such as the discrete Fourier transform (DFT) and the discrete Hartley transform (DHT) are important tools in numerical analysis. The successful application of transform techniques relies on the existence of efficient…
Gradient descent for matrix factorization exhibits an implicit bias toward approximately low-rank solutions. While existing theories often assume the boundedness of iterates, empirically the bias persists even with unbounded sequences. This…
Matrix factorization methods are important tools in data mining and analysis. They can be used for many tasks, ranging from dimensionality reduction to visualization. In this paper we concentrate on the use of matrix factorizations for…
Identifying discrete patterns in binary data is an important dimensionality reduction tool in machine learning and data mining. In this paper, we consider the problem of low-rank binary matrix factorisation (BMF) under Boolean arithmetic.…
The sparsity constrained rank-one matrix approximation problem is a difficult mathematical optimization problem which arises in a wide array of useful applications in engineering, machine learning and statistics, and the design of…
Boolean matrix factorization is a natural and a popular technique for summarizing binary matrices. In this paper, we study a problem of Boolean matrix factorization where we additionally require that the factor matrices have consecutive…
Integer factorization is a fundamental problem in algorithmic number theory and computer science. It is considered as a one way or trapdoor function in the (RSA) cryptosystem. To date, from elementary trial division to sophisticated methods…
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…