  A key question in many low-rank problems throughout optimization, machine
learning, and statistics is to characterize the convex hulls of simple low-rank
sets and judiciously apply these convex hulls to obtain strong yet
computationally tractable convex relaxations. We invoke the matrix perspective
function - the matrix analog of the perspective function - and characterize
explicitly the convex hull of epigraphs of simple matrix convex functions under
low-rank constraints. Further, we combine the matrix perspective function with
orthogonal projection matrices-the matrix analog of binary variables which
capture the row-space of a matrix-to develop a matrix perspective reformulation
technique that reliably obtains strong relaxations for a variety of low-rank
problems, including reduced rank regression, non-negative matrix factorization,
and factor analysis. Moreover, we establish that these relaxations can be
modeled via semidefinite constraints and thus optimized over tractably. The
proposed approach parallels and generalizes the perspective reformulation
technique in mixed-integer optimization and leads to new relaxations for a
broad class of problems.
