  Models specified by low-rank matrices are ubiquitous in contemporary
applications. In many of these problem domains, the row/column space structure
of a low-rank matrix carries information about some underlying phenomenon, and
it is of interest in inferential settings to evaluate the extent to which the
row/column spaces of an estimated low-rank matrix signify discoveries about the
phenomenon. However, in contrast to variable selection, we lack a formal
framework to assess true/false discoveries in low-rank estimation; in
particular, the key source of difficulty is that the standard notion of a
discovery is a discrete one that is ill-suited to the smooth structure
underlying low-rank matrices. We address this challenge via a geometric
reformulation of the concept of a discovery, which then enables a natural
definition in the low-rank case. We describe and analyze a generalization of
the Stability Selection method of Meinshausen and B\"uhlmann to control for
false discoveries in low-rank estimation, and we demonstrate its utility
compared to previous approaches via numerical experiments.
