Rank, decomposition, and uniqueness for 3-way and n-way arrays
Multiway data analysis
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
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The aim of this paper is the introduction of a new method for the numerical computation of the rank of a three-way array $\mathbf{X}\in\mathbb{R}^{I\times J\times K}$ over $\mathbb{R}$. We show that the rank of a three-way array over $\mathbb{R}$ is intimately related to the real solution set of a system of polynomial equations. Using this, we present some numerical results based on the computation of Gröbner bases. Also, we show that for $I=(K-1)(J-1)+1$ and $2\leq K\leq J\leq I$, the rank for generic data has more than one rank value, and the minimum attained value is $I$.