---
_id: '13315'
abstract:
- lang: eng
text: How do statistical dependencies in measurement noise influence high-dimensional
inference? To answer this, we study the paradigmatic spiked matrix model of principal
components analysis (PCA), where a rank-one matrix is corrupted by additive noise.
We go beyond the usual independence assumption on the noise entries, by drawing
the noise from a low-order polynomial orthogonal matrix ensemble. The resulting
noise correlations make the setting relevant for applications but analytically
challenging. We provide characterization of the Bayes optimal limits of inference
in this model. If the spike is rotation invariant, we show that standard spectral
PCA is optimal. However, for more general priors, both PCA and the existing approximate
message-passing algorithm (AMP) fall short of achieving the information-theoretic
limits, which we compute using the replica method from statistical physics. We
thus propose an AMP, inspired by the theory of adaptive Thouless–Anderson–Palmer
equations, which is empirically observed to saturate the conjectured theoretical
limit. This AMP comes with a rigorous state evolution analysis tracking its performance.
Although we focus on specific noise distributions, our methodology can be generalized
to a wide class of trace matrix ensembles at the cost of more involved expressions.
Finally, despite the seemingly strong assumption of rotation-invariant noise,
our theory empirically predicts algorithmic performance on real data, pointing
at strong universality properties.
acknowledgement: J.B. was funded by the European Union (ERC, CHORAL, project number
101039794). Views and opinions expressed are however those of the author(s) only
and do not necessarily reflect those of the European Union or the European Research
Council. Neither the European Union nor the granting authority can be held responsible
for them. M.M. was supported by the 2019 Lopez-Loreta Prize. We would like to thank
the reviewers for the insightful comments and, in particular, for suggesting the
BAMP-inspired denoisers leading to AMP-AP.
article_number: e2302028120
article_processing_charge: Yes (in subscription journal)
article_type: original
author:
- first_name: Jean
full_name: Barbier, Jean
last_name: Barbier
- first_name: Francesco
full_name: Camilli, Francesco
last_name: Camilli
- first_name: Marco
full_name: Mondelli, Marco
id: 27EB676C-8706-11E9-9510-7717E6697425
last_name: Mondelli
orcid: 0000-0002-3242-7020
- first_name: Manuel
full_name: Sáenz, Manuel
last_name: Sáenz
citation:
ama: Barbier J, Camilli F, Mondelli M, Sáenz M. Fundamental limits in structured
principal component analysis and how to reach them. Proceedings of the National
Academy of Sciences of the United States of America. 2023;120(30). doi:10.1073/pnas.2302028120
apa: Barbier, J., Camilli, F., Mondelli, M., & Sáenz, M. (2023). Fundamental
limits in structured principal component analysis and how to reach them. Proceedings
of the National Academy of Sciences of the United States of America. National
Academy of Sciences. https://doi.org/10.1073/pnas.2302028120
chicago: Barbier, Jean, Francesco Camilli, Marco Mondelli, and Manuel Sáenz. “Fundamental
Limits in Structured Principal Component Analysis and How to Reach Them.” Proceedings
of the National Academy of Sciences of the United States of America. National
Academy of Sciences, 2023. https://doi.org/10.1073/pnas.2302028120.
ieee: J. Barbier, F. Camilli, M. Mondelli, and M. Sáenz, “Fundamental limits in
structured principal component analysis and how to reach them,” Proceedings
of the National Academy of Sciences of the United States of America, vol.
120, no. 30. National Academy of Sciences, 2023.
ista: Barbier J, Camilli F, Mondelli M, Sáenz M. 2023. Fundamental limits in structured
principal component analysis and how to reach them. Proceedings of the National
Academy of Sciences of the United States of America. 120(30), e2302028120.
mla: Barbier, Jean, et al. “Fundamental Limits in Structured Principal Component
Analysis and How to Reach Them.” Proceedings of the National Academy of Sciences
of the United States of America, vol. 120, no. 30, e2302028120, National Academy
of Sciences, 2023, doi:10.1073/pnas.2302028120.
short: J. Barbier, F. Camilli, M. Mondelli, M. Sáenz, Proceedings of the National
Academy of Sciences of the United States of America 120 (2023).
date_created: 2023-07-30T22:01:02Z
date_published: 2023-07-25T00:00:00Z
date_updated: 2025-09-09T12:41:50Z
day: '25'
ddc:
- '000'
department:
- _id: MaMo
doi: 10.1073/pnas.2302028120
external_id:
isi:
- '001121663500001'
pmid:
- '37463204'
file:
- access_level: open_access
checksum: 1fc06228afdb3aa80cf8e7766bcf9dc5
content_type: application/pdf
creator: dernst
date_created: 2023-07-31T07:30:48Z
date_updated: 2023-07-31T07:30:48Z
file_id: '13323'
file_name: 2023_PNAS_Barbier.pdf
file_size: 995933
relation: main_file
success: 1
file_date_updated: 2023-07-31T07:30:48Z
has_accepted_license: '1'
intvolume: ' 120'
isi: 1
issue: '30'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
pmid: 1
project:
- _id: 059876FA-7A3F-11EA-A408-12923DDC885E
name: Prix Lopez-Loretta 2019 - Marco Mondelli
publication: Proceedings of the National Academy of Sciences of the United States
of America
publication_identifier:
eissn:
- 1091-6490
publication_status: published
publisher: National Academy of Sciences
quality_controlled: '1'
related_material:
link:
- relation: software
url: https://github.com/fcamilli95/Structured-PCA-
scopus_import: '1'
status: public
title: Fundamental limits in structured principal component analysis and how to reach
them
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 120
year: '2023'
...