  High-dimensional feature vectors are likely to contain sets of measurements
that are approximate replicates of one another. In complex applications, or
automated data collection, these feature sets are not known a priori, and need
to be determined. This work proposes a class of latent factor models on the
observed high-dimensional random vector $X \in \mathbb{R}^p$, for defining,
identifying and estimating the index set of its approximately replicate
components. The model class is parametrized by a $p \times K$ loading matrix
$A$ that contains a hidden sub-matrix whose rows can be partitioned into groups
of parallel vectors. Under this model class, a set of approximate replicate
components of $X$ corresponds to a set of parallel rows in $A$: these entries
of $X$ are, up to scale and additive error, the same linear combination of the
$K$ latent factors; the value of $K$ is itself unknown. The problem of finding
approximate replicates in $X$ reduces to identifying, and estimating, the
location of the hidden sub-matrix within $A$, and of the partition of its row
index set $H$. Both $H$ and its partiton can be fully characterized in terms of
a new family of criteria based on the correlation matrix of $X$, and their
identifiability, as well as that of the unknown latent dimension $K$, are
obtained as consequences. The constructive nature of the identifiability
arguments enables computationally efficient procedures, with consistency
guarantees. When $A$ has the errors-in-variable parametrization, the difficulty
of the problem is elevated. The task becomes that of separating out groups of
parallel rows that are proportional to canonical basis vectors from other dense
parallel rows in $A$. This is met under a scale assumption, via a principled
way of selecting the target row indices, guided by the succesive maximization
of Schur complements of appropriate covariance matrices.
