Year: 2026
Type: preprint
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 [12,10,16,24]. 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 [19]. 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.
Source: arXiv (Cornell University)
Institutions: Indian Statistical Institute, University of Chicago
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223062811