Projection Retrieval: Theory and Algorithms

We consider the fundamental problem of determining a low-rank orthogonal projection operator P from measurements of the form parallel to Px parallel to. 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.