Bridging Sparse Models and Factor Models

Check out the new "Bridging Factor and Sparse Models," by Fan, Masini, and Medeiros.  So many interesting connections to think about. 

A key distinction is hard constraints vs. soft constraints.  Both sparsity and factor structure (reduced rank) are examples of hard constraints, and in that sense they are more similar than different. Consider an  NxN parameter matrix M with elements a, b, ... Sparsity is the hard constraint M (or some subset of M) = 0, e.g., a = b = 0.  But one can of course consider more general hard constraints.  An obvious class is of the form  f(M) = 0, e.g., a + b^2 = 3.  Another obvious class, in multivariate environments, is reduced-rank (rank(M) < N) -- that is, factor structure.