  Integrated principal components analysis, or iPCA, is an unsupervised
learning technique for grouped vector data recently defined by Tang and Allen.
Like PCA, iPCA computes new axes that best explain the variance of the data,
but iPCA is designed to handle corrupting influences by the elements within
each group on one another - e.g. data about students at a school grouped into
classrooms. Tang and Allen showed empirically that regularized iPCA finds
useful features for such grouped data in practice. However, it is not yet known
when unregularized iPCA generically exists. For contrast, PCA (which is a
special case of iPCA) typically exists whenever the number of data points
exceeds the dimension. We study this question and find that the answer is
significantly more complicated than it is for PCA. Despite this complexity, we
find simple sufficient conditions for a very useful case - when the groups are
no more than half as large as the dimension and the total number of data points
exceeds the dimension, iPCA generically exists. We also fully characterize the
existence of iPCA in case all the groups are the same size. When all groups are
not the same size, however, we find that the group sizes for which iPCA
generically exists are the integral points in a non-convex union of polyhedral
cones. Nonetheless, we exhibit an algorithm to decide whether iPCA generically
exists that is polynomial time in the node dimensions (based on the affirmative
answer for the saturation conjecture by Knutson and Tao as well as a very
simple randomized algorithm.At its core, our approach identifies connections
between iPCA and stability notions for star quivers, thus bringing tools from
invariant theory and quiver representations to the table.
