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 $N \ge 2$ 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 $A_i\in\mathbb{R}^{m_i\times n}$ as $A_i=U_i\Sigma_i V^\text{T}$, but requires that each of the matrices $A_i$ has full column rank. We propose a modification 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 approach. We extend the notion of common subspaces to isolated subspaces, which identify features that are unique to one $A_i$. 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 with $m_i<n$ or $\text{rank}(A_i)<n$, such as are encountered in bioinformatics, neuroscience, control theory or classification problems.