Two new fixed polarity linear Kronecker transforms executed over GF(3) are introduced in this article. Both transforms are based on recursive equations using Kronecker products what allows to obtain simple corresponding fast transforms and very regular butterfly diagrams. The computational costs to calculate the new pair of transforms and their experimental comparison with ternary Reed-Muller t...

An analytical theory of power law graphs is presented based on the Kronecker graph generation technique. The analysis uses Kronecker exponentials of complete bipartite graphs to formulate the the substructure of such graphs. This allows various high level quantities (e.g. degree distribution, betweenness centrality, diameter, eigenvalues, and isoparametric ratio) to be computed directly from th...

Kronecker’s views on the foundations of mathematics are often reduced to jokes and regarded as an outdated set of ill-assorted ideas. A closer look however shows that they constitute an original and coherent doctrine justified by epistemological convictions. This doctrine appears in the article On the concept of number, published in the Journal of Crelle (1887) and, especially, in the last cour...

The second-order optimization methods, notably the D-KFAC (Distributed Kronecker Factored Approximate Curvature) algorithms, have gained traction on accelerating deep neural network (DNN) training GPU clusters. However, existing algorithms require to compute and communicate a large volume of information, i.e., factors (KFs), before preconditioning gradients, resulting in computation communicati...

Pairwise classification has many applications including network prediction, entity resolution, and collaborative filtering. The pairwise kernel has been proposed for those purposes by several research groups independently, and become successful in various fields. In this paper, we propose an efficient alternative which we call Cartesian kernel. While the existing pairwise kernel (which we refer...

The Kronecker product is a key algorithm and is ubiquitous across the physical, biological, and computation social sciences. Thus considerations of optimal implementation are important. The need to have high performance and computational reproducibility is paramount. Moreover, due to the need to compose multiple Kronecker products, issues related to data structures, layout and indexing algebra ...

Measurements taken at the campus of Brigham Young University (BYU) are used to investigate the statistical properties of the indoor MIMO channel. Two statistical tests, Royston’s and HenzeZirkler’s, are applied to the MIMO data to assess whether the data belongs to a multivariate normal distribution or not. The possibility of modeling the covariance matrix as a Kronecker product of the correlat...

How can we generate realistic graphs? In addition, how can we do so with a mathematically tractable model that makes it feasible to analyze their properties rigorously? Real graphs obey a long list of surprising properties: Heavy tails for the inand out-degree distribution; heavy tails for the eigenvalues and eigenvectors; small diameters; and the recently discovered “Densification Power Law” (...

This paper establishes a straightforward interconnection between the Kronecker canonical form and the special coordinate basis of linear systems. Such an interconnection enables the computation of the Kronecker canonical form, and as a by-product, the Smith form, of the system matrix of general multivariable time-invariant linear systems. The overall procedure involves the transformation of a g...

The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C T; A G], where C, T, A, G are the letters of the genetic alphabet. The matrix [C T; A G] in the second Kronecker power is the (4*4)-matrix of 16 duplets. The matrix [C T; A G] in the third Kronecker power is the (8*8)-matrix of 64 triplets. It is significant that peculiarities of the degeneracy...