Independent subspace analysis is unique, given irreducibility

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Abstract

Independent Subspace Analysis (ISA) is a generalization of ICA. It tries to find a basis in which a given random vector can be decomposed into groups of mutually independent random vectors. Since the first introduction of ISA, various algorithms to solve this problem have been introduced, however a general proof of the uniqueness of ISA decompositions remained an open question. In this contribution we address this question and sketch a proof for the separability of ISA. The key condition for separability is to require the subspaces to be not further decomposable (irreducible). Based on a decomposition into irreducible components, we formulate a general model for ISA without restrictions on the group sizes. The validity of the uniqueness result is illustrated on a toy example. Moreover, an extension of ISA to subspace extraction is introduced and its indeterminacies are discussed. © Springer-Verlag Berlin Heidelberg 2007.

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Gutch, H. W., & Theis, F. J. (2007). Independent subspace analysis is unique, given irreducibility. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 4666 LNCS, pp. 49–56). Springer Verlag. https://doi.org/10.1007/978-3-540-74494-8_7

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