Practically all methods for efficient computation with multidimensional models take advantage of the fact that the model in question in a way factorizes. It means that it is possible to decompose the model into its low-dimensional parts, each of which can be defined with a reasonable number of parameters. This is a basic idea of computation with probabilistic Graphical Markov Models as well as ...

We continue the study of the so-called thematic factorizations of admissible very badly approximable matrix functions. These factorizations were introduced by V.V. Peller and N.J. Young for studying superoptimal approximation by bounded analytic matrix functions. Even though thematic indices associated with a thematic factorization of an admissible very badly approximable matrix function are no...

Many of the currently popular ‘block algorithms’ are scalar algorithms in which the operations have been grouped and reordered into matrix operations. One genuine block algorithm in practical use is block LU factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown here that block LU factorization is stable if A is block diagonally dominant by c...

In spite of their great success, traditional factorization algorithms typically do not support features (e.g., Matrix Factorization), or their complexity scales quadratically with the number of features (e.g, Factorization Machine). On the other hand, neural methods allow large feature sets, but are often designed for a specific application. We propose novel deep factorization methods that allo...

A (d, d + 1)-graph is a graph whose vertices all have degrees in the set {d, d + 1}. Such a graph is semiregular. An (r, r + 1)-factorization of a graph G is a decomposition of G into (r, r + 1)-factors. For d-regular simple graphs G we say for which x and r G must have an (r, r + 1)-factorization with exactly x (r, r + 1)-factors. We give similar results for (d, d + 1)-simple graphs and for (d...

We first present a model-independent analysis using flavor SU(3) symmetry with SU(3) breaking effects in B decays to two charmless vector mesons (V V ) in the final state. In order to bridge the flavor SU(3) symmetry approach and the factorization approach in B → V V decays, we explicitly show how to translate each SU(3) amplitude into a corresponding amplitude in factorization. Various decay m...

We introduce the strong recursive skeletonization factorization (RS-S), a new approximate matrix factorization based on recursive skeletonization for solving discretizations of linear integral equations associated with elliptic partial differential equations in two and three dimensions (and other matrices with similar hierarchical rank structure). Unlike previous skeletonizationbased factorizat...

An error analysis result is given for classical Gram–Schmidt factorization of a full rank matrix A into A = QR where Q is left orthogonal (has orthonormal columns) and R is upper triangular. The work presented here shows that the computed R satisfies R R = A A + E where E is an appropriately small backward error, but only if the diagonals of R are computed in a manner similar to Cholesky factor...

We introduce the strong recursive skeletonization factorization (RS-S), a new approximate matrix factorization based on recursive skeletonization for solving discretizations of linear integral equations associated with elliptic partial differential equations in two and three dimensions (and other matrices with similar hierarchical rank structure). Unlike previous skeletonizationbased factorizat...

Steady state simulations in Computational Fluid Dynamics (CFD), which rely on implicit time integration, are not experiencing great accelerations on GPUs. Moreover, most of the reported acceleration effort concerns solving the linear system of equations while neglecting the acceleration potential of running the entire simulation on the GPU. In this paper, we present the software implementation ...