In this paper we prove a new characterization of the max-plus singular values of a maxplus matrix, as the max-plus eigenvalues of an associated max-plus matrix pencil. This new characterization allows us to compute max-plus singular values quickly and accurately. As well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially parameterized we show experimentally that max-plus singular values give order of magnitude approximations to the classical singular values of parameter independent classical matrices.We also discuss Hungarian scaling, which is a diagonal scaling strategy for preprocessing classical linear systems. We show that Hungarian scaling can dramatically reduce the d-norm condition number and that this action can be explained using our new theory for max-plus singular values.
Abstract. We present a new method for the a priori approximation of the orders of magnitude of the entries in the LU factors of a complex or real matrix A. This approximation is used to determine the positions of the largest entries in the LU factors of A, and these positions are used as the sparsity pattern for an incomplete LU factorization preconditioner. Our method uses max-plus algebra and is based solely on the moduli of the entries of A. We also present techniques for predicting which permutation matrices will be chosen by Gaussian elimination with partial pivoting. We exploit the strong connection between the field of Puiseux series and the max-plus semiring to prove properties of the max-plus LU factors. Experiments with a set of test matrices from the University of Florida Sparse Matrix Collection show that our max-plus LU preconditioners outperform traditional level of fill methods and have similar performance to those preconditioners computed with more expensive threshold-based methods.Key words. max-plus algebra, LU factorization, Hungarian scaling, linear systems of equations, sparse matrices, incomplete LU factorization, preconditioning AMS subject classifications. 65F08, 65F30, 15A23, 15A80 DOI. 10.1137/16M10945791. Introduction. Max-plus algebra is the analogue of linear algebra developed for the binary operations max and plus over the real numbers together with −∞, the latter playing the role of additive identity. Max-plus algebraic techniques have already been used in numerical linear algebra to, for example, approximate the orders of magnitude of the roots of scalar polynomials [19], approximate the moduli of the eigenvalues of matrix polynomials [1,10,14], and approximate singular values [9]. These approximations have been used as starting points for iterative schemes and in the design of preprocessing steps to improve the numerical stability of standard algorithms [3,6,7,14,20]. Our aim is to show how max-plus algebra can be used to approximate the sizes of the entries in the LU factors of a complex or real matrix A and how these approximations can subsequently be used in the construction of an incomplete LU (ILU) factorization preconditioner for A.In order to be able to apply max-plus techniques to the matrix A ∈ C n×n we must first transform it into a max-plus matrix. We do this using the valuation map
Nonnegative matrix factorization (NMF) is one of the most frequently-used matrix factorization models in data analysis. A significant reason to the popularity of NMF is its interpretability and the 'parts of whole' interpretation of its components. Recently, max-times, or subtropical, matrix factorization (SMF) has been introduced as an alternative model with equally interpretable 'winner takes it all' interpretation. In this paper we propose a new mixed linear-tropical model, and a new algorithm, called Latitude, that combines NMF and SMF, being able to smoothly alternate between the two. In our model, the data is modeled using the latent factors and latent parameters that control whether the factors are interpreted as NMF or SMF features, or their mixtures. We present an algorithm for our novel matrix factorization. Our experiments show that our algorithm improves over both baselines, and can yield interpretable results that reveal more of the latent structure than either NMF or SMF alone.
The directed acyclic graph (DAG) associated with a parallel algorithm captures the partial order in which separate local computations are completed and how their outputs are subsequently used in further computations. Unlike in a synchronous parallel algorithm the DAG associated with an asynchronous parallel algorithm is not predetermined. Instead it is a product of the asynchronous timing dynamics of the machine and cannot be known in advance, as such it is best thought of as a pseudorandom variable.In this paper we present a formalism for analyzing the performance of asynchronous parallel Jacobi's method in terms of its DAG. We use this approach to prove error bounds and bounds on the rate of convergence. The rate of convergence bounds are based on the statistical properties of the DAG and are valid for systems with a non-negative iteration matrix. We support our theoretical results with a suit of numerical examples, where we compare the performance of synchronous and asynchronous parallel Jacobi to certain statistical properties of the DAGs associated with the computations. We also present some examples of small matrices with elements of mixed sign, which demonstrate that determining whether a system will converge under asynchronous iteration in this more general setting is a far more difficult problem.
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