Algebraic factor analysis: tetrads, pentads and beyond

By Mathias Drton, Bernd Sturmfels, Seth Sullivant. 2005

105 citations recorded by the source corpus.

Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a multivariate normal distribution. The parameter space of a factor analysis model is a subset of the cone of positive definite matrices. This parameter space is studied from the perspective of computational algebraic geometry. Gröbner bases and resultants are applied to compute the ideal of all polynomial functions that vanish on the parameter space. These polynomials, known as model invariants, arise from rank conditions on a symmetric matrix under elimination of the diagonal entries of the matrix. Besides revealing the geometry of the factor analysis model, the model invariants also furnish useful statistics for testing goodness-of-fit.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c69c4239bd7ca17871a853e12cf3236480a7c653

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Citations in the corpus

Noetherianity up to Symmetry (2013, unknown): “General stabilisation results for this model were ﬁrst conjectured in [ DSS07 ], and for 1 factor established prior to that in [ dLST95 ].”
Tensor decompositions for learning latent variable models (2012, background): “The relevance of tensor analysis to latent variable modeling has been long recognized in the field of algebraic statistics (Pachter and Sturmfels, 2005), and many works characterize the algebraic varieties corresponding to the moments of various classes of latent variable models (Drton et al., 2007;”
Moments of minors of Wishart matrices (2006, background): “2(ii) applies in particular to factor analysis with m− 1 factors; see also [6].”
A Bayesian information criterion for singular models (2013, background): “Note that the ‘top coefficient’ λii equals the dimension of the set of covariance matrices in model Mi; for a computation of this dimension see, for example, theorem 2 in Drton et al. (2007).”
Geometry of the Restricted Boltzmann Machine (2009, methodology): “ch is a subset Mk n of the probability simplex 2n 1. The graphical model for Gaussian random variables represented by Figure 1 is the factor analysis model, whose algebraic properties were studied in [3, 12]. Thus, the restricted Boltzmann machine is the binary undirected analog of factor analysis. ”
Finite generation of symmetric ideals (2004, background): “This result is motivated by finiteness questions in chemistry [10, 16, 17] and algebraic statistics [ 4 ] involving chains of invariant ideals Ik (k = 1,2,...) inside finite-dimensional polynomial rings Rk. Section 5 contains a discussion.”