Abstract
We decompose a matrix Y into a sum of bilinear terms in a stepwise manner, by considering Y as a mapping between two finite dimensional Banach spaces. We provide transition formulas, and represent them in a duality diagram, thus generalizing the well known duality diagram in the french school of data analysis. As an application, we introduce a family of Euclidean multidimensional scaling models.
Highlights
Matrix factorization, named decomposition, in data analysis is at the core of factor analysis; and one of its principal aims, as clearly stated by Hubert et al (2000), is to visualize geometrically the statistical association existing among the rows or the columns of the matrix
Singular value decomposition (SVD) is the most used matrix decomposition method in statistics; the aim of this paper is to present in a coherent way the theory of SVD-like matrix factorizations based on subordinate or induced norms; and at the same time, review the existing literature
We embedded the ordinary SVD into a larger family based on induced matrix norms, and provided the transition formulas and a simple criss-cross iterative procedure to compute the principal axes and principal factor scores
Summary
Matrix factorization, named decomposition, in data analysis is at the core of factor analysis; and one of its principal aims, as clearly stated by Hubert et al (2000), is to visualize geometrically the statistical association existing among the rows or the columns of the matrix. Singular value decomposition (SVD) is the most used matrix decomposition method in statistics; the aim of this paper is to present in a coherent way the theory of SVD-like matrix factorizations based on subordinate or induced norms; and at the same time, review the existing literature. This presentation generalizes the SVD by embedding it in a larger family: It belongs to the class of optimal biconjugate decompositions; biconjugate decompositions are based on Wedderburn rank-one reduction theorem as described by Chu et al (1995). The rows or the columns of A can be used as starting values for a or b
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