Papers › A Higher-Order Generalized Singular Value Decomposition for Rank Deficient Matrices
A Higher-Order Generalized Singular Value Decomposition for Rank Deficient Matrices
Idris Kempf, Paul J. Goulart, Stephen R. Duncan
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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 Aᵢ∈ℝ^(mᵢ×n) as Aᵢ=UᵢΣᵢ Vᵀ, 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 datasets with with mᵢ<n or rank(Aᵢ)<n, such as are encountered in bioinformatics, neuroscience, control theory or classification problems.
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