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Uniqueness of linear factorizations into independent subspaces

Author

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  • Gutch, Harold W.
  • Theis, Fabian J.

Abstract

Given a random vector X, we address the question of linear separability of X, that is, the task of finding a linear operator W such that we have (S1,…,SM)=(WX) with statistically independent random vectors Si. As this requirement alone is already fulfilled trivially by X being independent of the empty rest, we require that the components be not further decomposable. We show that if X has finite covariance, such a representation is unique up to trivial indeterminacies. We propose an algorithm based on this proof and demonstrate its applicability. Related algorithms, however with fixed dimensionality of the subspaces, have already been successfully employed in biomedical applications, such as separation of fMRI recorded data. Based on the presented uniqueness result, it is now clear that also subspace dimensions can be determined in a unique and therefore meaningful fashion, which shows the advantages of independent subspace analysis in contrast to methods like principal component analysis.

Suggested Citation

  • Gutch, Harold W. & Theis, Fabian J., 2012. "Uniqueness of linear factorizations into independent subspaces," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 48-62.
    • Handle: RePEc:eee:jmvana:v:112:y:2012:i:c:p:48-62
    DOI: 10.1016/j.jmva.2012.05.019
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    References listed on IDEAS

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    1. Blanchard, Gilles & Kawanabe, Motoaki & Sugiyama, Masashi & Spokoiny, Vladimir & Müller, Klaus-Robert, 2006. "In search of non-Gaussian components of a high-dimensional distribution," SFB 649 Discussion Papers 2006-040, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
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