Metraipolis › archive › dc_040dfd6f9b4a

Geometric Fluctuations of the $\sinΘ$ Distance in High-Dimensional Principal Subspace Estimation — arXiv:2609.07751

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.07751
License (as archived): CC0-1.0
Archived: 2026-10-16T14:30:09.913Z
SHA-256 of the archived text: aae31e3790ad9548131c9ecfa349632a3e8ba0dadd91ebd67711839daba7e119
Stable URL of this page: https://metraipolis.com/documento/dc_040dfd6f9b4a

How to cite

Geometric Fluctuations of the $\sinΘ$ Distance in High-Dimensional Principal Subspace Estimation — arXiv:2609.07751 — arXiv (en), https://arxiv.org/abs/2609.07751 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_040dfd6f9b4a (Metraipolis archive dc_040dfd6f9b4a, sha256 aae31e3790ad9548131c9ecfa349632a3e8ba0dadd91ebd67711839daba7e119)

BibTeX

@misc{dc040dfd6f9b4a,
  title = {Geometric Fluctuations of the $sinΘ$ Distance in High-Dimensional Principal Subspace Estimation — arXiv:2609.07751},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.07751},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_040dfd6f9b4a; sha256 aae31e3790ad9548131c9ecfa349632a3e8ba0dadd91ebd67711839daba7e119},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.07751. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We investigate the geometric fluctuations of principal subspaces for high-dimensional covariance matrices through the squared Frobenius $\sin\Theta$ distance between the sample and population eigenspaces associated with the $r_p$ largest eigenvalues. An explicit first-order expansion and a central limit theorem are established for this subspace distance. The theory allows the subspace dimension to diverge subject to $r_p=o(n)$, where $n$ is the sample size. It also permits a diverging spectral norm of the population covariance matrix, population spikes of different orders, and repeated or closely spaced spikes. This sharp characterisation captures features of the subspace estimation error that are not reflected in existing perturbation bounds. As applications, we derive an explicit asymptotic expansion for the expected PCA excess risk and a refined error bound for distributed PCA. In both cases, existing upper bounds can increase with the spiked-block condition number when some leading spikes become stronger, whereas our results show that the corresponding estimation errors need not increase and may instead decrease. Numerical experiments reproduce this contrasting behaviour and demonstrate the finite-sample accuracy of our theoretical findings.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.ST).
Categorie arXiv: Statistics Theory (math.ST).
Pagina del lavoro: https://arxiv.org/abs/2609.07751 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:09:40.571Z dalla pagina https://arxiv.org/catchup/math.ST/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).