To lie or not to lie in a subspace
  Give deterministic necessary and sufficient conditions to guarantee that if a subspace fits certain partially observed data from a union of subspaces, it is because such data really lies in a subspace.   Furthermore, Give deterministic necessary and sufficient conditions to guarantee that if a subspace fits certain partially observed data, such subspace is unique.   Do this by characterizing when and only when a set of incomplete vectors behaves as a single but complete one. 
Daniel L. Pimentel-Alarc\'on
stat.ML cs.LG