  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 $A_i=U_i\Sigma_iV^{\text{T}}$, 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.
