A Central Limit Theorem for Intrinsic Fréchet Means on Real Grassmannians
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 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 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 Gr(1,2) for which the mean is unique but the Fréchet Hessian vanishes.
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BibTeX
@misc{banerjee2026central,
title = {{A Central Limit Theorem for Intrinsic Fréchet Means on Real Grassmannians}},
author = {Bipayan Banerjee and Imon Banerjee and Swagatam Das},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24159},
eprint = {2610.24159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24159},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Central Limit Theorem for Intrinsic Fréchet Means on Real Grassmannians
AU - Bipayan Banerjee
AU - Imon Banerjee
AU - Swagatam Das
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24159
UR - https://arxiv.org/abs/2610.24159
AB - 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 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 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 Gr(1,2) for which the mean is unique but the Fréchet Hessian vanishes.
ER -