Direct transformations yielding the knight's move pattern in 3x3x3 arrays

Three-way arrays (or tensors) can be regarded as extensions of the traditional two-way data matrices that have a third dimension. Studying algebraic properties of arrays is relevant, for example, for the Tucker three-way PCA method, which generalizes principal component analysis to three-way data. One important algebraic property of arrays is concerned with the possibility of transformations to simplicity. An array is said to be transformed to a simple form when it can be manipulated by a sequence of invertible operations such that a vast majority of its entries become zero. This paper shows how 3 × 3 × 3 arrays, whether symmetric or nonsymmetric, can be transformed to a simple form with 18 out of its 27 entries equal to zero. We call this simple form the “knight's move pattern” due to a loose resemblance to the moves of a knight in a game of chess. The pattern was examined by Kiers, Ten Berge, and Rocci. It will be shown how the knight's move pattern can be found by means of a numeric–algebraic procedure based on the Gröbner basis. This approach seems to work almost surely for randomly generated arrays, whether symmetric or nonsymmetric

Polonaise

https://digitalcommons.library.umaine.edu/mmb-ps/3188/thumbnail.jp

First and second-order derivatives for CP and INDSCAL

In this paper we provide the means to analyse the second-order differential structure of optimization functions concerning CANDECOMP/PARAFAC and INDSCAL. Closed-form formulas are given under two types of constraint: unit-length columns or orthonormality of two of the three component matrices. Some numerical problems that might occur during the computation of the Jacobian and Hessian matrices are addressed. The use of these matrices is illustrated in three applications. (C) 2010 Elsevier B.V. All rights reserved

Simplicity transformations for three-way arrays with symmetric slices, and applications to Tucker-3 models with sparse core arrays

AbstractTucker three-way PCA and Candecomp/Parafac are two well-known methods of generalizing principal component analysis to three way data. Candecomp/Parafac yields component matrices A (e.g., for subjects or objects), B (e.g., for variables) and C (e.g., for occasions) that are typically unique up to jointly permuting and rescaling columns. Tucker-3 analysis, on the other hand, has full transformational freedom. That is, the fit does not change when A,B, and C are postmultiplied by nonsingular transformation matrices, provided that the inverse transformations are applied to the so-called core array G̲. This freedom of transformation can be used to create a simple structure in A,B,C, and/or in G̲. This paper deals with the latter possibility exclusively. It revolves around the question of how a core array, or, in fact, any three-way array can be transformed to have a maximum number of zero elements. Direct applications are in Tucker-3 analysis, where simplicity of the core may facilitate the interpretation of a Tucker-3 solution, and in constrained Tucker-3 analysis, where hypotheses involving sparse cores are taken into account. In the latter cases, it is important to know what degree of sparseness can be attained as a tautology, by using the transformational freedom. In addition, simplicity transformations have proven useful as a mathematical tool to examine rank and generic or typical rank of three-way arrays. So far, a number of simplicity results have been attained, pertaining to arrays sampled randomly from continuous distributions. These results do not apply to three-way arrays with symmetric slices in one direction. The present paper offers a number of simplicity results for arrays with symmetric slices of order 2×2,3×3 and 4×4. Some generalizations to higher orders are also discussed. As a mathematical application, the problem of determining the typical rank of 4×3×3 and 5×3×3 arrays with symmetric slices will be revisited, using a sparse form with only 8 out of 36 elements nonzero for the former case and 10 out of 45 elements nonzero for the latter one, that can be attained almost surely for such arrays. The issue of maximal simplicity of the targets to be presented will be addressed, either by formal proofs or by relying on simulation results

Determinants of expression of SARS-CoV-2 entry-related genes in upper and lower airways.

Funder: Dutch Research Council (NWO)Funder: Cancer Research UK Cambridge CentreFunder: ATS Foundation/Boehringer Ingelheim Pharmaceuticals Inc. Research FellowshipFunder: The Netherlands Ministry of Spatial Planning, Housing, and the EnvironmentFunder: Chan Zuckerberg InitiativeFunder: The Netherlands Ministry of Health, Welfare, and SportFunder: Longfonds Junior FellowshipFunder: Cambridge BioresourceFunder: The Netherlands Organization for Health Research and DevelopmentFunder: Cambridge NIHR Biomedical Research CentreFunder: Parker B. Francis FellowshipFunder: China Scholarship Counci