Thin Hessenberg Pairs

A square matrix is called {\it Hessenberg} whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$ denote a nonzero finite-dimensional vector space over a field \fld. We consider an ordered pair of linear transformations $A: V \to V$ and $A^*: V \to V$ which satisfy both (i), (ii) below. \begin{enumerate} \item There exists a basis for $V$ with respect to which the matrix representing $A$ is Hessenberg and the matrix representing $A^*$ is diagonal. \item There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is Hessenberg. \end{enumerate} \noindent We call such a pair a {\it thin Hessenberg pair} (or {\it TH pair}). This is a special case of a {\it Hessenberg pair} which was introduced by the author in an earlier paper. We investigate several bases for $V$ with respect to which the matrices representing $A$ and $A^*$ are attractive. We display these matrices along with the transition matrices relating the bases. We introduce an "oriented" version of $A,A^*$ called a TH system. We classify the TH systems up to isomorphism.Comment: 23 page

Alternative methods for representing the inverse of linear programming basis matrices

Methods for representing the inverse of Linear Programming (LP) basis matrices are closely related to techniques for solving a system of sparse unsymmetric linear equations by direct methods. It is now well accepted that for these problems the static process of reordering the matrix in the lower block triangular (LBT) form constitutes the initial step. We introduce a combined static and dynamic factorisation of a basis matrix and derive its inverse which we call the partial elimination form of the inverse (PEFI). This factorization takes advantage of the LBT structure and produces a sparser representation of the inverse than the elimination form of the inverse (EFI). In this we make use of the original columns (of the constraint matrix) which are in the basis. To represent the factored inverse it is, however, necessary to introduce special data structures which are used in the forward and the backward transformations (the two major algorithmic steps) of the simplex method. These correspond to solving a system of equations and solving a system of equations with the transposed matrix respectively. In this paper we compare the nonzero build up of PEFI with that of EFI. We have also investigated alternative methods for updating the basis inverse in the PEFI representation. The results of our experimental investigation are presented in this pape

Rooted Trees Searching for Cocyclic Hadamard Matrices over D4t

A new reduction on the size of the search space for cocyclic Hadamard matrices over dihedral groups D4t is described, in terms of the so called central distribution. This new search space adopt the form of a forest consisting of two rooted trees (the vertices representing subsets of coboundaries) which contains all cocyclic Hadamard matrices satisfying the constraining condition. Experimental calculations indicate that the ratio between the number of constrained cocyclic Hadamard matrices and the size of the constrained search space is greater than the usual ratio.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM–296Junta de Andalucía P07-FQM-0298

RedisGraph GraphBLAS Enabled Graph Database

RedisGraph is a Redis module developed by Redis Labs to add graph database functionality to the Redis database. RedisGraph represents connected data as adjacency matrices. By representing the data as sparse matrices and employing the power of GraphBLAS (a highly optimized library for sparse matrix operations), RedisGraph delivers a fast and efficient way to store, manage and process graphs. Initial benchmarks indicate that RedisGraph is significantly faster than comparable graph databases.Comment: Accepted to IEEE IPDPS 2019 GrAPL worksho

Log-Euclidean Bag of Words for Human Action Recognition

Representing videos by densely extracted local space-time features has recently become a popular approach for analysing actions. In this paper, we tackle the problem of categorising human actions by devising Bag of Words (BoW) models based on covariance matrices of spatio-temporal features, with the features formed from histograms of optical flow. Since covariance matrices form a special type of Riemannian manifold, the space of Symmetric Positive Definite (SPD) matrices, non-Euclidean geometry should be taken into account while discriminating between covariance matrices. To this end, we propose to embed SPD manifolds to Euclidean spaces via a diffeomorphism and extend the BoW approach to its Riemannian version. The proposed BoW approach takes into account the manifold geometry of SPD matrices during the generation of the codebook and histograms. Experiments on challenging human action datasets show that the proposed method obtains notable improvements in discrimination accuracy, in comparison to several state-of-the-art methods