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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
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
matrices
as
but requires that each of the matrices
has full column rank. We propose a modification of the HO-GSVD that extends its applicability to rank-deficient data matrices
. If the matrix of stacked
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
. 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
or rank
, such as those encountered in bioinformatics, neuroscience, control theory, and classification problems.
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