Projection retrieval: Theory and algorithms

We consider the fundamental problem of determining a low-rank orthogonal projection operator P from measurements of the form || Px||. First, we leverage a nonembedding result for the complex Grassmannian to establish and analyze a lower bound on the number of measurements necessary to uniquely determine every possible P. Next, we provide a collection of particularly few measurement vectors that uniquely determine almost every P. Finally, we propose manifold-constrained least-squares optimization as a general technique for projection retrieval.