A Central Limit Theorem for Intrinsic Fréchet Means on Real Grassmannians

Source abstract

Subspace-valued data arise in dimension reduction, image-set and video recognition, and signal processing, and the Fréchet mean is their standard sample mean [11,9,15,23]. We establish a central limit theorem for the intrinsic sample Fréchet mean of i.i.d. observations on the real Grassmann manifold $\mathrm{Gr}(p,n)$, as a specialization of the general theorem of [18]. Unlike prior results which require the distribution to vanish or place negligible mass near the cut locus of the population mean, we allow a continuous, possibly positive density. To enable our results, we derive two properties of $\mathrm{Gr}(p,n)$: the first is that the union of the cut loci of all points within distance $δ$ of the mean has volume $O(δ)$. The second is that the Hessian of the squared distance function is integrable. We also give a smooth density on $\mathrm{Gr}(1,2)$ for which the mean is unique but the Fréchet Hessian vanishes.