The selective regularization of a linear regression model

This paper discusses constructing a linear regression model with regularization of the system matrix of normal equations. In contrast to the conventional ridge regression, where positive parameters are added to all diagonal terms of a matrix, in the method proposed only those matrix diagonal entries that correspond to the data with a high correlation are increased. This leads to a decrease in the matrix conditioning and, therefore, to a decrease in the corresponding coefficients of the regression equation. The selection of the entries to be increased is based on the triangular decomposition of the correlation matrix of the original dataset. The effectiveness of the method is tested on a known dataset, and it is performed not only with a ridge regression, but also with the results of applying the widespread algorithms LARS and Lasso.

Decomposition of Goursat Matrices and Subgroups of Zm x Zn

Given the number of subgroups of Zm x Zn, we deduce the Goursat matrix. The purpose of this paper is two-fold. A first and more concrete aim is to demonstrate that the triangular decomposition of the Goursat matrix may also be written out explicitly, and furthermore that the same is true of the inverse of these triangular factors. A second and more abstract aim provides a containment relation property between subgroups of a direct product . Namely, if U2 ≤ U1 ≤ Zm x Zn, we provide necessary and sufficient conditions for U2 ≤ U1.

MPI+OpenMP parallel implementation of conjugate gradient method with preconditioner of block partial inverse triangular decomposition of IC2S and IC1

The paper proposes a new preconditioner for solving systems of linear algebraic equations with a symmetric positively defined matrix by the method of conjugate gradients – Block Incomplete Inverse Cholesky BIIC preconditioner in combination with a triangular first-order decomposition "by value" - BIIC-IC1. The algorithm based on MPI+OpenMP techniques is proposed for the construction and application of the BIIC preconditioner combined with stabilized triangular decomposition of the second order "by value" (BIIC-IS2S). In this case, the BIIC-IC2S preconditioner uses the number of blocks multiple of the number of processors used and the number of threads used. Two algorithms based on MPI+OpenMP techniques are proposed for the construction and application of the BIIC-IC1 preconditioner. Comparative timing results for the MPI+OpenMP and MPI implementations of the proposed preconditioning used with the conjugate gradient method for a model problem and the sparse matrix collections SuiteSparse are presented.

Modules for loop Affine-Virasoro algebras

In this paper, we study the representations of loop Affine-Virasoro algebras. As they have canonical triangular decomposition, we define Verma modules and their irreducible quotients. We give necessary and sufficient condition for a irreducible highest weight module to have finite dimensional weight spaces. We prove that an irreducible integrable module is either a highest weight module or a lowest weight module whenever the canonical central element acts non-trivially. At the end, we construct Affine central operators for each integer and they commute with the action of the Affine Lie algebra.