---
OA_place: publisher
OA_type: diamond
PlanS_conform: '1'
_id: '20734'
abstract:
- lang: eng
  text: We consider the problem of parameter estimation in a high-dimensional generalized
    linear model. Spectral methods obtained via the principal eigenvector of a suitable
    data-dependent matrix provide a simple yet surprisingly effective solution. However,
    despite their wide use, a rigorous performance characterization, as well as a
    principled way to preprocess the data, are available only for unstructured (i.i.d.
    Gaussian and Haar orthogonal) designs. In contrast, real-world data matrices are
    highly structured and exhibit non-trivial correlations. To address the problem,
    we consider correlated Gaussian designs capturing the anisotropic nature of the
    features via a covariance matrix Σ. Our main result is a precise asymptotic characterization
    of the performance of spectral estimators. This allows us to identify the optimal
    preprocessing that minimizes the number of samples needed for parameter estimation.
    Surprisingly, such preprocessing is universal across a broad set of designs, which
    partly addresses a conjecture on optimal spectral estimators for rotationally
    invariant models. Our principled approach vastly improves upon previous heuristic
    methods, including for designs common in computational imaging and genetics. The
    proposed methodology, based on approximate message passing, is broadly applicable
    and opens the way to the precise characterization of spiked matrices and of the
    corresponding spectral methods in a variety of settings.
acknowledgement: "This work was done when Y. Z. and H. C. J. were at the Institute
  of Science and Technology Austria. Y. Z. thanks Hugo Latourelle-Vigeant for bringing
  [53] to the authors’ attention.\r\nY. Z. and M. M. are partially supported by the
  2019 Lopez-Loreta Prize and by the Interdisciplinary Projects Committee (IPC) at
  ISTA. H. C. J. is supported by the ERC Advanced Grant “RMTBeyond” No. 101020331."
article_processing_charge: No
article_type: original
author:
- first_name: Yihan
  full_name: Zhang, Yihan
  id: 2ce5da42-b2ea-11eb-bba5-9f264e9d002c
  last_name: Zhang
  orcid: 0000-0002-6465-6258
- first_name: Hong Chang
  full_name: Ji, Hong Chang
  last_name: Ji
- first_name: Ramji
  full_name: Venkataramanan, Ramji
  last_name: Venkataramanan
- first_name: Marco
  full_name: Mondelli, Marco
  id: 27EB676C-8706-11E9-9510-7717E6697425
  last_name: Mondelli
  orcid: 0000-0002-3242-7020
citation:
  ama: Zhang Y, Ji HC, Venkataramanan R, Mondelli M. Spectral estimators for structured
    generalized linear models via approximate message passing. <i>Mathematical Statistics
    and Learning</i>. 2025;8(3-4):193-304. doi:<a href="https://doi.org/10.4171/MSL/52">10.4171/MSL/52</a>
  apa: Zhang, Y., Ji, H. C., Venkataramanan, R., &#38; Mondelli, M. (2025). Spectral
    estimators for structured generalized linear models via approximate message passing.
    <i>Mathematical Statistics and Learning</i>. EMS Press. <a href="https://doi.org/10.4171/MSL/52">https://doi.org/10.4171/MSL/52</a>
  chicago: Zhang, Yihan, Hong Chang Ji, Ramji Venkataramanan, and Marco Mondelli.
    “Spectral Estimators for Structured Generalized Linear Models via Approximate
    Message Passing.” <i>Mathematical Statistics and Learning</i>. EMS Press, 2025.
    <a href="https://doi.org/10.4171/MSL/52">https://doi.org/10.4171/MSL/52</a>.
  ieee: Y. Zhang, H. C. Ji, R. Venkataramanan, and M. Mondelli, “Spectral estimators
    for structured generalized linear models via approximate message passing,” <i>Mathematical
    Statistics and Learning</i>, vol. 8, no. 3–4. EMS Press, pp. 193–304, 2025.
  ista: Zhang Y, Ji HC, Venkataramanan R, Mondelli M. 2025. Spectral estimators for
    structured generalized linear models via approximate message passing. Mathematical
    Statistics and Learning. 8(3–4), 193–304.
  mla: Zhang, Yihan, et al. “Spectral Estimators for Structured Generalized Linear
    Models via Approximate Message Passing.” <i>Mathematical Statistics and Learning</i>,
    vol. 8, no. 3–4, EMS Press, 2025, pp. 193–304, doi:<a href="https://doi.org/10.4171/MSL/52">10.4171/MSL/52</a>.
  short: Y. Zhang, H.C. Ji, R. Venkataramanan, M. Mondelli, Mathematical Statistics
    and Learning 8 (2025) 193–304.
corr_author: '1'
date_created: 2025-12-07T23:02:02Z
date_published: 2025-09-02T00:00:00Z
date_updated: 2025-12-09T13:53:31Z
day: '02'
ddc:
- '000'
department:
- _id: MaMo
doi: 10.4171/MSL/52
file:
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  creator: dernst
  date_created: 2025-12-09T13:50:03Z
  date_updated: 2025-12-09T13:50:03Z
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has_accepted_license: '1'
intvolume: '         8'
issue: 3-4
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 193-304
project:
- _id: 059876FA-7A3F-11EA-A408-12923DDC885E
  name: Prix Lopez-Loretta 2019 - Marco Mondelli
publication: Mathematical Statistics and Learning
publication_identifier:
  eissn:
  - 2520-2324
  issn:
  - 2520-2316
publication_status: published
publisher: EMS Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Spectral estimators for structured generalized linear models via approximate
  message passing
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2025'
...