What to Expect When You Are Expecting on the Grassmannian
  Consider an incoming sequence of vectors, all belonging to an unknown subspace $\operatorname{S}$, and each with many missing entries. In order to estimate $\operatorname{S}$, it is common to partition the data into blocks and iteratively update the estimate of $\operatorname{S}$ with each new incoming measurement block.   In this paper, we investigate a rather basic question: Is it possible to identify $\operatorname{S}$ by averaging the column span of the partially observed incoming measurement blocks on the Grassmannian?   We show that in general the span of the incoming blocks is in fact a biased estimator of $\operatorname{S}$ when data suffers from erasures, and we find an upper bound for this bias. We reach this conclusion by examining the defining optimization program for the Fr\'{e}chet expectation on the Grassmannian, and with the aid of a sharp perturbation bound and standard large deviation results. 
Armin Eftekhari, Laura Balzano, and Michael B. Wakin
cs.IT math.IT