---
_id: '10211'
abstract:
- lang: eng
  text: "We study the problem of recovering an unknown signal \U0001D465\U0001D465
    given measurements obtained from a generalized linear model with a Gaussian sensing
    matrix. Two popular solutions are based on a linear estimator \U0001D465\U0001D465^L
    and a spectral estimator \U0001D465\U0001D465^s. The former is a data-dependent
    linear combination of the columns of the measurement matrix, and its analysis
    is quite simple. The latter is the principal eigenvector of a data-dependent matrix,
    and a recent line of work has studied its performance. In this paper, we show
    how to optimally combine \U0001D465\U0001D465^L and \U0001D465\U0001D465^s. At
    the heart of our analysis is the exact characterization of the empirical joint
    distribution of (\U0001D465\U0001D465,\U0001D465\U0001D465^L,\U0001D465\U0001D465^s)
    in the high-dimensional limit. This allows us to compute the Bayes-optimal combination
    of \U0001D465\U0001D465^L and \U0001D465\U0001D465^s, given the limiting distribution
    of the signal \U0001D465\U0001D465. When the distribution of the signal is Gaussian,
    then the Bayes-optimal combination has the form \U0001D703\U0001D465\U0001D465^L+\U0001D465\U0001D465^s
    and we derive the optimal combination coefficient. In order to establish the limiting
    distribution of (\U0001D465\U0001D465,\U0001D465\U0001D465^L,\U0001D465\U0001D465^s),
    we design and analyze an approximate message passing algorithm whose iterates
    give \U0001D465\U0001D465^L and approach \U0001D465\U0001D465^s. Numerical simulations
    demonstrate the improvement of the proposed combination with respect to the two
    methods considered separately."
acknowledgement: M. Mondelli would like to thank Andrea Montanari for helpful discussions.
  All the authors would like to thank the anonymous reviewers for their helpful comments.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Marco
  full_name: Mondelli, Marco
  id: 27EB676C-8706-11E9-9510-7717E6697425
  last_name: Mondelli
  orcid: 0000-0002-3242-7020
- first_name: Christos
  full_name: Thrampoulidis, Christos
  last_name: Thrampoulidis
- first_name: Ramji
  full_name: Venkataramanan, Ramji
  last_name: Venkataramanan
citation:
  ama: Mondelli M, Thrampoulidis C, Venkataramanan R. Optimal combination of linear
    and spectral estimators for generalized linear models. <i>Foundations of Computational
    Mathematics</i>. 2022;22(5):1513-1566. doi:<a href="https://doi.org/10.1007/s10208-021-09531-x">10.1007/s10208-021-09531-x</a>
  apa: Mondelli, M., Thrampoulidis, C., &#38; Venkataramanan, R. (2022). Optimal combination
    of linear and spectral estimators for generalized linear models. <i>Foundations
    of Computational Mathematics</i>. Springer. <a href="https://doi.org/10.1007/s10208-021-09531-x">https://doi.org/10.1007/s10208-021-09531-x</a>
  chicago: Mondelli, Marco, Christos Thrampoulidis, and Ramji Venkataramanan. “Optimal
    Combination of Linear and Spectral Estimators for Generalized Linear Models.”
    <i>Foundations of Computational Mathematics</i>. Springer, 2022. <a href="https://doi.org/10.1007/s10208-021-09531-x">https://doi.org/10.1007/s10208-021-09531-x</a>.
  ieee: M. Mondelli, C. Thrampoulidis, and R. Venkataramanan, “Optimal combination
    of linear and spectral estimators for generalized linear models,” <i>Foundations
    of Computational Mathematics</i>, vol. 22, no. 5. Springer, pp. 1513–1566, 2022.
  ista: Mondelli M, Thrampoulidis C, Venkataramanan R. 2022. Optimal combination of
    linear and spectral estimators for generalized linear models. Foundations of Computational
    Mathematics. 22(5), 1513–1566.
  mla: Mondelli, Marco, et al. “Optimal Combination of Linear and Spectral Estimators
    for Generalized Linear Models.” <i>Foundations of Computational Mathematics</i>,
    vol. 22, no. 5, Springer, 2022, pp. 1513–66, doi:<a href="https://doi.org/10.1007/s10208-021-09531-x">10.1007/s10208-021-09531-x</a>.
  short: M. Mondelli, C. Thrampoulidis, R. Venkataramanan, Foundations of Computational
    Mathematics 22 (2022) 1513–1566.
date_created: 2021-11-03T10:59:08Z
date_published: 2022-10-01T00:00:00Z
date_updated: 2025-04-15T06:53:08Z
day: '01'
ddc:
- '510'
department:
- _id: MaMo
doi: 10.1007/s10208-021-09531-x
external_id:
  arxiv:
  - '2008.03326'
  isi:
  - '000685721000001'
file:
- access_level: open_access
  checksum: 9ea12dd8045a0678000a3a59295221cb
  content_type: application/pdf
  creator: alisjak
  date_created: 2021-12-13T15:47:54Z
  date_updated: 2021-12-13T15:47:54Z
  file_id: '10542'
  file_name: 2021_Springer_Mondelli.pdf
  file_size: 2305731
  relation: main_file
  success: 1
file_date_updated: 2021-12-13T15:47:54Z
has_accepted_license: '1'
intvolume: '        22'
isi: 1
issue: '5'
keyword:
- Applied Mathematics
- Computational Theory and Mathematics
- Computational Mathematics
- Analysis
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '10'
oa: 1
oa_version: Published Version
page: 1513-1566
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Foundations of Computational Mathematics
publication_identifier:
  eissn:
  - 1615-3383
  issn:
  - 1615-3375
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
status: public
title: Optimal combination of linear and spectral estimators for 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: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 22
year: '2022'
...