A higher-order generalized singular value decomposition for rank-deficient matrices

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 common subspaces that are shared across multiple large-scale datasets with different row dimensions. The standard HO-GSVD factors N matrices Ai ∈ ℝmi×n as Ai = Ui∑iVT but requires that each of the matrices Ai has full column rank. We propose a modification of the HO-GSVD that extends its applicability to rank-deficient data matrices Ai. If the matrix of stacked Ai has full rank, we show that the properties of the original HO-GSVD extend to our approach. We extend the notion of common subspaces to isolated subspaces, which identify features that are unique to one Ai. 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 matrices with mi < n or rank (Ai) < n, such as those encountered in bioinformatics, neuroscience, control theory, and classification problems