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Cover It Up! Bipartite Graphs Uncover Identifiability in Sparse Factor Analysis

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  • Darjus Hosszejni
  • Sylvia Fruhwirth-Schnatter

Abstract

Despite the popularity of factor models with sparse loading matrices, little attention has been given to formally address identifiability of these models beyond standard rotation-based identification such as the positive lower triangular constraint. To fill this gap, we present a counting rule on the number of nonzero factor loadings that is sufficient for achieving generic uniqueness of the variance decomposition in the factor representation. This is formalized in the framework of sparse matrix spaces and some classical elements from graph and network theory. Furthermore, we provide a computationally efficient tool for verifying the counting rule. Our methodology is illustrated for real data in the context of post-processing posterior draws in Bayesian sparse factor analysis.

Suggested Citation

  • Darjus Hosszejni & Sylvia Fruhwirth-Schnatter, 2022. "Cover It Up! Bipartite Graphs Uncover Identifiability in Sparse Factor Analysis," Papers 2211.00671, arXiv.org, revised Feb 2025.
    • Handle: RePEc:arx:papers:2211.00671
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    1. repec:bfi:wpaper:2014-014 is not listed on IDEAS
    2. Conti, Gabriella & Frühwirth-Schnatter, Sylvia & Heckman, James J. & Piatek, Rémi, 2014. "Bayesian exploratory factor analysis," Journal of Econometrics, Elsevier, vol. 183(1), pages 31-57.
    3. Henry Kaiser, 1958. "The varimax criterion for analytic rotation in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 23(3), pages 187-200, September.
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    5. Kaufmann, Sylvia & Schumacher, Christian, 2019. "Bayesian estimation of sparse dynamic factor models with order-independent and ex-post mode identification," Journal of Econometrics, Elsevier, vol. 210(1), pages 116-134.
    6. Benjamin Williams, 2020. "Identification of the linear factor model," Econometric Reviews, Taylor & Francis Journals, vol. 39(1), pages 92-109, January.
    7. Sirio Legramanti & Daniele Durante & David B Dunson, 2020. "Bayesian cumulative shrinkage for infinite factorizations," Biometrika, Biometrika Trust, vol. 107(3), pages 745-752.
    8. A. Bhattacharya & D. B. Dunson, 2011. "Sparse Bayesian infinite factor models," Biometrika, Biometrika Trust, vol. 98(2), pages 291-306.
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    Cited by:

    1. Sylvia Frühwirth-Schnatter & Darjus Hosszejni & Hedibert Freitas Lopes, 2023. "When It Counts—Econometric Identification of the Basic Factor Model Based on GLT Structures," Econometrics, MDPI, vol. 11(4), pages 1-30, November.
    2. Sylvia Fruhwirth-Schnatter, 2023. "Generalized Cumulative Shrinkage Process Priors with Applications to Sparse Bayesian Factor Analysis," Papers 2303.00473, arXiv.org.

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