  We consider the problem of extracting a low-dimensional, linear latent
variable structure from high-dimensional random variables. Specifically, we
show that under mild conditions and when this structure manifests itself as a
linear space that spans the conditional means, it is possible to consistently
recover the structure using only information up to the second moments of these
random variables. This finding, specialized to one-parameter exponential
families whose variance function is quadratic in their means, allows for the
derivation of an explicit estimator of such latent structure. This approach
serves as a latent variable model estimator and as a tool for dimension
reduction for a high-dimensional matrix of data composed of many related
variables. Our theoretical results are verified by simulation studies and an
application to genomic data.
