  Various applications in signal processing and machine learning give rise to
highly structured spectral optimization problems characterized by low-rank
solutions. Two important examples that motivate this work are optimization
problems from phase retrieval and from blind deconvolution, which are designed
to yield rank-1 solutions. An algorithm is described that is based on solving a
certain constrained eigenvalue optimization problem that corresponds to the
gauge dual which, unlike the more typical Lagrange dual, has an especially
simple constraint. The dominant cost at each iteration is the computation of
rightmost eigenpairs of a Hermitian operator. A range of numerical examples
illustrate the scalability of the approach.
