  The Trek Separation Theorem (Sullivant et al. 2010) states necessary and
sufficient conditions for a linear directed acyclic graphical model to entail
for all possible values of its linear coefficients that the rank of various
sub-matrices of the covariance matrix is less than or equal to n, for any given
n. In this paper, I extend the Trek Separation Theorem in two ways: I prove
that the same necessary and sufficient conditions apply even when the
generating model is partially non-linear and contains some cycles. This
justifies application of constraint-based causal search algorithms such as the
BuildPureClusters algorithm (Silva et al. 2006) for discovering the causal
structure of latent variable models to data generated by a wider class of
causal models that may contain non-linear and cyclic relations among the latent
variables.

Artificial Intelligence;Machine Learning