A Higher-Order Generalized Singular Value Decomposition for Rank Deficient Matrices

02/19/2021 ∙ by Idris Kempf, et al. ∙ 0

The higher-order generalized singular value decomposition (HO-GSVD) is a matrix factorization technique that extends the GSVD to N ≥ 2 data matrices, and can be used to identify shared subspaces in multiple large-scale datasets with different row dimensions. The standard HO-GSVD factors N matrices A_i∈ℝ^m_i× n as A_i=U_iΣ_iV^T, but requires that each of the matrices A_i has full column rank. We propose a reformulation of the HO-GSVD that extends its applicability to rank-deficient data matrices A_i. If the matrix of stacked A_i has full rank, we show that the properties of the original HO-GSVD extend to our reformulation. The HO-GSVD captures shared right singular vectors of the matrices A_i, and we show that our method also identifies directions that are unique to the image of a single matrix. We also extend our results to the higher-order cosine-sine decomposition (HO-CSD), which is closely related to the HO-GSVD. Our extension of the standard HO-GSVD allows its application to datasets with m_i < n, such as are encountered in bioinformatics, neuroscience, control theory or classification problems.

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