{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-higher-order-generalized-singular-value","title":"A Higher-Order Generalized Singular Value Decomposition for Rank Deficient Matrices","arxiv_id":"2102.09822","date":"2021-02-19","proceeding":null,"authors":["Idris Kempf","Paul J. Goulart","Stephen R. Duncan"],"abstract":"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