---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '22228'
abstract:
- lang: eng
  text: "In a mixed generalized linear model, the goal is to learn multiple signals
    from unlabeled observations: each sample comes from exactly one signal, but it
    is not known which one. We consider the prototypical problem of estimating two
    statistically independent signals in a mixed generalized linear model with Gaussian
    covariates. Spectral methods are a popular class of estimators which output the
    top two eigenvectors of a suitable data-dependent matrix. However, despite the
    wide applicability, their design is still obtained via heuristic considerations,
    and the number of samples \U0001D45B needed to guarantee recovery is superlinear
    in the signal dimension \U0001D451. In this paper, we develop exact asymptotics
    on spectral methods in the challenging proportional regime in which \U0001D45B,\U0001D451
    grow large and their ratio converges to a finite constant. This allows us optimize
    the design of the spectral method, and combine it with a simple linear estimator,
    to minimize the estimation error. Our characterization exploits a mix of tools
    from random matrices, free probability, and the theory of approximate message
    passing algorithms. Numerical simulations for mixed linear regression and phase
    retrieval demonstrate the advantage enabled by our analysis over existing designs
    of spectral methods."
acknowledgement: The first and second authors were partially supported by the 2019
  Lopez-Loreta prize.
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Yihan
  full_name: Zhang, Yihan
  last_name: Zhang
- first_name: Marco
  full_name: Mondelli, Marco
  id: 27EB676C-8706-11E9-9510-7717E6697425
  last_name: Mondelli
  orcid: 0000-0002-3242-7020
- first_name: Ramji
  full_name: Venkataramanan, Ramji
  last_name: Venkataramanan
citation:
  ama: Zhang Y, Mondelli M, Venkataramanan R. Precise asymptotics for spectral methods
    in mixed generalized linear models. <i>SIAM Journal on Mathematics of Data Science</i>.
    2026;8(2):411-439. doi:<a href="https://doi.org/10.1137/24m1702854">10.1137/24m1702854</a>
  apa: Zhang, Y., Mondelli, M., &#38; Venkataramanan, R. (2026). Precise asymptotics
    for spectral methods in mixed generalized linear models. <i>SIAM Journal on Mathematics
    of Data Science</i>. Society for Industrial &#38; Applied Mathematics. <a href="https://doi.org/10.1137/24m1702854">https://doi.org/10.1137/24m1702854</a>
  chicago: Zhang, Yihan, Marco Mondelli, and Ramji Venkataramanan. “Precise Asymptotics
    for Spectral Methods in Mixed Generalized Linear Models.” <i>SIAM Journal on Mathematics
    of Data Science</i>. Society for Industrial &#38; Applied Mathematics, 2026. <a
    href="https://doi.org/10.1137/24m1702854">https://doi.org/10.1137/24m1702854</a>.
  ieee: Y. Zhang, M. Mondelli, and R. Venkataramanan, “Precise asymptotics for spectral
    methods in mixed generalized linear models,” <i>SIAM Journal on Mathematics of
    Data Science</i>, vol. 8, no. 2. Society for Industrial &#38; Applied Mathematics,
    pp. 411–439, 2026.
  ista: Zhang Y, Mondelli M, Venkataramanan R. 2026. Precise asymptotics for spectral
    methods in mixed generalized linear models. SIAM Journal on Mathematics of Data
    Science. 8(2), 411–439.
  mla: Zhang, Yihan, et al. “Precise Asymptotics for Spectral Methods in Mixed Generalized
    Linear Models.” <i>SIAM Journal on Mathematics of Data Science</i>, vol. 8, no.
    2, Society for Industrial &#38; Applied Mathematics, 2026, pp. 411–39, doi:<a
    href="https://doi.org/10.1137/24m1702854">10.1137/24m1702854</a>.
  short: Y. Zhang, M. Mondelli, R. Venkataramanan, SIAM Journal on Mathematics of
    Data Science 8 (2026) 411–439.
corr_author: '1'
das_tickbox: '0'
date_created: 2026-06-30T13:03:41Z
date_published: 2026-06-01T00:00:00Z
date_updated: 2026-07-01T06:29:52Z
day: '01'
ddc:
- '000'
department:
- _id: MaMo
doi: 10.1137/24m1702854
external_id:
  arxiv:
  - '2211.11368'
file:
- access_level: open_access
  checksum: 5cfd350dc64d1476063e959316dbff65
  content_type: application/pdf
  creator: dernst
  date_created: 2026-07-01T06:22:15Z
  date_updated: 2026-07-01T06:22:15Z
  file_id: '22230'
  file_name: 2026_SIAMJourmathDataScience_Zhang.pdf
  file_size: 1210346
  relation: main_file
  success: 1
file_date_updated: 2026-07-01T06:22:15Z
has_accepted_license: '1'
intvolume: '         8'
issue: '2'
keyword:
- spectral estimator
- generalized linear models
- mixed regression
- high-dimensional asymptotics
- random matrix theory
- approximate message passing (AMP)
language:
- iso: eng
mathsc:
- '62E20'
- 62J05
- 62J12
month: '06'
oa: 1
oa_version: Published Version
page: 411-439
project:
- _id: 059876FA-7A3F-11EA-A408-12923DDC885E
  name: Prix Lopez-Loretta 2019 - Marco Mondelli
publication: SIAM Journal on Mathematics of Data Science
publication_identifier:
  eissn:
  - 2577-0187
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Precise asymptotics for spectral methods in mixed generalized linear models
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: '2026'
...