---
references:
- id: Akimoto2017Fast
  accessed:
    - year: 2023
      month: 4
      day: 7
  author:
    - family: Akimoto
      given: Youhei
  citation-key: Akimoto2017Fast
  DOI: 10.48550/arXiv.1706.02069
  issued:
    - year: 2017
      month: 6
      day: 7
  number: arXiv:1706.02069
  publisher: arXiv
  title: Fast Eigen Decomposition for Low-Rank Matrix Approximation
  type: article
  URL: http://arxiv.org/abs/1706.02069

- id: Ameli2023Singular
  accessed:
    - year: 2023
      month: 4
      day: 26
  author:
    - family: Ameli
      given: Siavash
    - family: Shadden
      given: Shawn C.
  citation-key: Ameli2023Singular
  container-title: Applied Mathematics and Computation
  container-title-short: Applied Mathematics and Computation
  DOI: 10.1016/j.amc.2023.128032
  ISSN: '00963003'
  issued:
    - year: 2023
      month: 4
  language: en
  page: '128032'
  title: >-
    A Singular Woodbury and Pseudo-Determinant Matrix Identities and Application
    to Gaussian Process Regression
  type: article-journal
  URL: http://arxiv.org/abs/2207.08038
  volume: '452'

- id: Babacan2012Sparse
  accessed:
    - year: 2022
      month: 2
      day: 4
  author:
    - family: Babacan
      given: S. Derin
    - family: Luessi
      given: Martin
    - family: Molina
      given: Rafael
    - family: Katsaggelos
      given: Aggelos K.
  citation-key: Babacan2012Sparse
  container-title: IEEE Transactions on Signal Processing
  container-title-short: IEEE Trans. Signal Process.
  DOI: 10.1109/TSP.2012.2197748
  ISSN: 1053-587X, 1941-0476
  issue: '8'
  issued:
    - year: 2012
      month: 8
  language: en
  page: 3964-3977
  title: Sparse Bayesian Methods for Low-Rank Matrix Estimation
  type: article-journal
  volume: '60'

- id: Bach2013Sharp
  accessed:
    - year: 2016
      month: 6
      day: 20
  author:
    - family: Bach
      given: Francis R.
  citation-key: Bach2013Sharp
  container-title: COLT
  issued:
    - year: 2013
  page: 185–209
  title: Sharp analysis of low-rank kernel matrix approximations.
  type: paper-conference
  URL: http://www.jmlr.org/proceedings/papers/v30/Bach13.pdf
  volume: '30'

- id: Bach2019Randomized
  accessed:
    - year: 2023
      month: 4
      day: 8
  author:
    - family: Bach
      given: C
    - family: Ceglia
      given: D.
    - family: Song
      given: L.
    - family: Duddeck
      given: F.
  citation-key: Bach2019Randomized
  container-title: International Journal for Numerical Methods in Engineering
  DOI: 10.1002/nme.6009
  ISSN: 1097-0207
  issue: '4'
  issued:
    - year: 2019
  language: en
  page: 209-241
  title: >-
    Randomized low-rank approximation methods for projection-based model order
    reduction of large nonlinear dynamical problems
  type: article-journal
  volume: '118'

- id: Bamieh2022Tutorial
  accessed:
    - year: 2023
      month: 11
      day: 26
  author:
    - family: Bamieh
      given: Bassam
  citation-key: Bamieh2022Tutorial
  DOI: 10.48550/arXiv.2002.05001
  issued:
    - year: 2022
      month: 4
      day: 24
  number: arXiv:2002.05001
  publisher: arXiv
  title: A Tutorial on Matrix Perturbation Theory (using compact matrix notation)
  type: article
  URL: http://arxiv.org/abs/2002.05001

- id: Barbier2017Layered
  accessed:
    - year: 2018
      month: 1
      day: 15
  author:
    - family: Barbier
      given: Jean
    - family: Macris
      given: Nicolas
    - family: Miolane
      given: Léo
  citation-key: Barbier2017Layered
  container-title: arXiv:1709.10368 [cond-mat, physics:math-ph]
  issued:
    - year: 2017
      month: 9
      day: 29
  title: The Layered Structure of Tensor Estimation and its Mutual Information
  type: article-journal
  URL: http://arxiv.org/abs/1709.10368

- id: Bauckhage2015KMeans
  accessed:
    - year: 2016
      month: 1
      day: 4
  author:
    - family: Bauckhage
      given: Christian
  citation-key: Bauckhage2015KMeans
  container-title: arXiv:1512.07548 [stat]
  issued:
    - year: 2015
      month: 12
      day: 23
  title: k-Means Clustering Is Matrix Factorization
  type: article-journal
  URL: http://arxiv.org/abs/1512.07548

- id: Berry2007Algorithms
  accessed:
    - year: 2015
      month: 5
      day: 31
  author:
    - family: Berry
      given: Michael W.
    - family: Browne
      given: Murray
    - family: Langville
      given: Amy N.
    - family: Pauca
      given: V. Paul
    - family: Plemmons
      given: Robert J.
  citation-key: Berry2007Algorithms
  container-title: Computational Statistics & Data Analysis
  container-title-short: Computational Statistics & Data Analysis
  DOI: 10.1016/j.csda.2006.11.006
  ISSN: 0167-9473
  issue: '1'
  issued:
    - year: 2007
      month: 9
      day: 15
  page: 155-173
  title: Algorithms and applications for approximate nonnegative matrix factorization
  type: article-journal
  URL: http://users.wfu.edu/plemmons/papers/BBLPP-rev.pdf
  volume: '52'

- id: Brand2002Incremental
  accessed:
    - year: 2023
      month: 4
      day: 8
  author:
    - family: Brand
      given: Matthew
  citation-key: Brand2002Incremental
  collection-editor:
    - family: Goos
      given: Gerhard
    - family: Hartmanis
      given: Juris
    - family: Leeuwen
      given: Jan
      non-dropping-particle: van
  container-title: Computer Vision — ECCV 2002
  DOI: 10.1007/3-540-47969-4_47
  editor:
    - family: Heyden
      given: Anders
    - family: Sparr
      given: Gunnar
    - family: Nielsen
      given: Mads
    - family: Johansen
      given: Peter
  event-place: Berlin, Heidelberg
  ISBN: 978-3-540-43745-1 978-3-540-47969-7
  issued:
    - year: 2002
  language: en
  page: 707-720
  publisher: Springer Berlin Heidelberg
  publisher-place: Berlin, Heidelberg
  title: >-
    Incremental Singular Value Decomposition of Uncertain Data with Missing
    Values
  type: paper-conference
  URL: https://www.merl.com/publications/docs/TR2002-24.pdf
  volume: '2350'

- id: Brand2006Fast
  accessed:
    - year: 2023
      month: 4
      day: 8
  author:
    - family: Brand
      given: Matthew
  citation-key: Brand2006Fast
  collection-title: Special Issue on Large Scale Linear and Nonlinear Eigenvalue Problems
  container-title: Linear Algebra and its Applications
  container-title-short: Linear Algebra and its Applications
  DOI: 10.1016/j.laa.2005.07.021
  ISSN: 0024-3795
  issue: '1'
  issued:
    - year: 2006
      month: 5
      day: 1
  language: en
  page: 20-30
  title: Fast low-rank modifications of the thin singular value decomposition
  type: article-journal
  volume: '415'

- id: Chen2018Harnessing
  author:
    - family: Chen
      given: Yudong
    - family: Chi
      given: Yuejie
  citation-key: Chen2018Harnessing
  container-title: IEEE Signal Processing Magazine
  DOI: 10.1109/MSP.2018.2821706
  ISSN: 1558-0792
  issue: '4'
  issued:
    - year: 2018
      month: 7
  page: 14-31
  title: >-
    Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation:
    Recent Theory and Fast Algorithms via Convex and Nonconvex Optimization
  type: article-journal
  URL: http://arxiv.org/abs/1802.08397
  volume: '35'

- id: Chi2019Nonconvex
  accessed:
    - year: 2019
      month: 11
      day: 26
  author:
    - family: Chi
      given: Yuejie
    - family: Lu
      given: Yue M.
    - family: Chen
      given: Yuxin
  citation-key: Chi2019Nonconvex
  container-title: IEEE Transactions on Signal Processing
  container-title-short: IEEE Trans. Signal Process.
  DOI: 10.1109/TSP.2019.2937282
  ISSN: 1053-587X, 1941-0476
  issue: '20'
  issued:
    - year: 2019
      month: 10
      day: 15
  page: 5239-5269
  title: 'Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview'
  type: article-journal
  URL: http://arxiv.org/abs/1809.09573
  volume: '67'

- id: Cichocki2016LowRank
  accessed:
    - year: 2016
      month: 9
      day: 9
  author:
    - family: Cichocki
      given: A.
    - family: Lee
      given: N.
    - family: Oseledets
      given: I. V.
    - family: Phan
      given: A.-H.
    - family: Zhao
      given: Q.
    - family: Mandic
      given: D.
  citation-key: Cichocki2016LowRank
  container-title: arXiv:1609.00893 [cs]
  issued:
    - year: 2016
      month: 9
      day: 4
  title: >-
    Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale
    Optimization Problems: Perspectives and Challenges PART 1
  type: article-journal
  URL: http://arxiv.org/abs/1609.00893

- id: Dahleh1990Matrix
  accessed:
    - year: 2023
      month: 11
      day: 26
  author:
    - family: Dahleh
      given: Mohammed
    - family: Dahleh
      given: Munther A.
    - family: Verghese
      given: George
  citation-key: Dahleh1990Matrix
  container-title: Lectures on Dynamic Systems and Control
  issued:
    - year: 1990
      month: 6
  language: en
  page: '91'
  title: Matrix Perturbations
  type: chapter
  URL: >-
    https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/855766ddb3a22777d5425518c0ad51d2_MIT6_241JS11_chap05.pdf
  volume: '20'

- id: Drineas2005Nystrom
  accessed:
    - year: 2014
      month: 10
      day: 10
  author:
    - family: Drineas
      given: Petros
    - family: Mahoney
      given: Michael W.
  citation-key: Drineas2005Nystrom
  container-title: Journal of Machine Learning Research
  ISSN: 1532-4435
  issued:
    - year: 2005
      month: 12
  page: 2153–2175
  title: >-
    On the Nyström method for approximating a Gram matrix for improved
    kernel-based learning
  type: article-journal
  URL: http://jmlr.org/papers/volume6/drineas05a/drineas05a.pdf
  volume: '6'

- id: Fasi2023Computing
  accessed:
    - year: 2023
      month: 4
      day: 17
  author:
    - family: Fasi
      given: Massimiliano
    - family: Higham
      given: Nicholas J.
    - family: Liu
      given: Xiaobo
  citation-key: Fasi2023Computing
  container-title: SIAM Journal on Matrix Analysis and Applications
  container-title-short: SIAM J. Matrix Anal. Appl.
  DOI: 10.1137/22M1471559
  ISSN: 0895-4798, 1095-7162
  issue: '1'
  issued:
    - year: 2023
      month: 3
      day: 31
  language: en
  page: 156-174
  title: >-
    Computing the Square Root of a Low-Rank Perturbation of the Scaled Identity
    Matrix
  type: article-journal
  volume: '44'

- id: Flammia2012Quantum
  accessed:
    - year: 2016
      month: 5
      day: 13
  author:
    - family: Flammia
      given: Steven T.
    - family: Gross
      given: David
    - family: Liu
      given: Yi-Kai
    - family: Eisert
      given: Jens
  citation-key: Flammia2012Quantum
  container-title: New Journal of Physics
  DOI: 10.1088/1367-2630/14/9/095022
  ISSN: 1367-2630
  issue: '9'
  issued:
    - year: 2012
      month: 9
      day: 27
  page: '095022'
  title: >-
    Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity,
    and Efficient Estimators
  type: article-journal
  URL: http://arxiv.org/abs/1205.2300
  volume: '14'

- id: Ghashami2015Frequent
  accessed:
    - year: 2016
      month: 6
      day: 24
  author:
    - family: Ghashami
      given: Mina
    - family: Liberty
      given: Edo
    - family: Phillips
      given: Jeff M.
    - family: Woodruff
      given: David P.
  citation-key: Ghashami2015Frequent
  container-title: arXiv:1501.01711 [cs]
  issued:
    - year: 2015
      month: 1
      day: 7
  title: 'Frequent Directions : Simple and Deterministic Matrix Sketching'
  type: article-journal
  URL: http://arxiv.org/abs/1501.01711

- id: Gordon2002Generalized2
  accessed:
    - year: 2022
      month: 5
      day: 13
  author:
    - family: Gordon
      given: Geoffrey J.
  citation-key: Gordon2002Generalized2
  collection-title: NIPS'02
  container-title: >-
    Proceedings of the 15th International Conference on Neural Information
    Processing Systems
  event-place: Cambridge, MA, USA
  issued:
    - year: 2002
      month: 1
      day: 1
  page: 593–600
  publisher: MIT Press
  publisher-place: Cambridge, MA, USA
  title: Generalized² Linear² models
  type: paper-conference
  URL: >-
    http://www.cs.cmu.edu/afs/cs.cmu.edu/project/ri-seminar/www/archives/2004.spring/ggllm.pdf

- id: Gross2010Quantum
  accessed:
    - year: 2016
      month: 5
      day: 13
  author:
    - family: Gross
      given: David
    - family: Liu
      given: Yi-Kai
    - family: Flammia
      given: Steven T.
    - family: Becker
      given: Stephen
    - family: Eisert
      given: Jens
  citation-key: Gross2010Quantum
  container-title: Physical Review Letters
  DOI: 10.1103/PhysRevLett.105.150401
  ISSN: 0031-9007, 1079-7114
  issue: '15'
  issued:
    - year: 2010
      month: 10
      day: 4
  title: Quantum state tomography via compressed sensing
  type: article-journal
  URL: http://arxiv.org/abs/0909.3304
  volume: '105'

- id: Gross2011Recovering
  author:
    - family: Gross
      given: D.
  citation-key: Gross2011Recovering
  container-title: IEEE Transactions on Information Theory
  DOI: 10.1109/TIT.2011.2104999
  ISSN: 0018-9448
  issue: '3'
  issued:
    - year: 2011
      month: 3
  page: 1548-1566
  title: Recovering Low-Rank Matrices From Few Coefficients in Any Basis
  type: article-journal
  URL: http://arxiv.org/abs/0910.1879
  volume: '57'

- id: Hager1989Updating
  accessed:
    - year: 2023
      month: 5
      day: 16
  author:
    - family: Hager
      given: William W.
  citation-key: Hager1989Updating
  container-title: SIAM Review
  container-title-short: SIAM Rev.
  DOI: 10.1137/1031049
  ISSN: 0036-1445
  issue: '2'
  issued:
    - year: 1989
      month: 6
  page: 221-239
  publisher: Society for Industrial and Applied Mathematics
  title: Updating the Inverse of a Matrix
  type: article-journal
  URL: >-
    https://web.archive.org/web/20170810045028id_/http://users.clas.ufl.edu/hager/papers/lightning/update.pdf
  volume: '31'

- id: Halko2010Finding
  accessed:
    - year: 2022
      month: 10
      day: 22
  author:
    - family: Halko
      given: Nathan
    - family: Martinsson
      given: Per-Gunnar
    - family: Tropp
      given: Joel A.
  citation-key: Halko2010Finding
  DOI: 10.48550/arXiv.0909.4061
  issued:
    - year: 2010
      month: 12
      day: 14
  number: arXiv:0909.4061
  publisher: arXiv
  title: >-
    Finding structure with randomness: Probabilistic algorithms for constructing
    approximate matrix decompositions
  type: article
  URL: http://arxiv.org/abs/0909.4061

- id: Harbrecht2012Lowrank
  accessed:
    - year: 2023
      month: 2
      day: 26
  author:
    - family: Harbrecht
      given: Helmut
    - family: Peters
      given: Michael
    - family: Schneider
      given: Reinhold
  citation-key: Harbrecht2012Lowrank
  collection-title: >-
    Third Chilean Workshop on Numerical Analysis of Partial Differential
    Equations (WONAPDE 2010)
  container-title: Applied Numerical Mathematics
  container-title-short: Applied Numerical Mathematics
  DOI: 10.1016/j.apnum.2011.10.001
  ISSN: 0168-9274
  issue: '4'
  issued:
    - year: 2012
      month: 4
      day: 1
  language: en
  page: 428-440
  title: On the low-rank approximation by the pivoted Cholesky decomposition
  type: article-journal
  volume: '62'

- id: Harville1976Extension
  accessed:
    - year: 2023
      month: 4
      day: 26
  author:
    - family: Harville
      given: David A.
  citation-key: Harville1976Extension
  container-title: The Annals of Statistics
  DOI: 10.1214/aos/1176343414
  ISSN: 0090-5364, 2168-8966
  issue: '2'
  issued:
    - year: 1976
      month: 3
  page: 384-395
  publisher: Institute of Mathematical Statistics
  title: >-
    Extension of the Gauss-Markov Theorem to Include the Estimation of Random
    Effects
  type: article-journal
  volume: '4'

- id: Harville1977Maximum
  accessed:
    - year: 2023
      month: 4
      day: 26
  author:
    - family: Harville
      given: David A.
  citation-key: Harville1977Maximum
  container-title: Journal of the American Statistical Association
  DOI: 10.1080/01621459.1977.10480998
  ISSN: 0162-1459
  issue: '358'
  issued:
    - year: 1977
      month: 6
      day: 1
  page: 320-338
  publisher: Taylor & Francis
  title: >-
    Maximum Likelihood Approaches to Variance Component Estimation and to
    Related Problems
  type: article-journal
  volume: '72'

- id: Hastie2015Matrix
  accessed:
    - year: 2023
      month: 4
      day: 20
  author:
    - family: Hastie
      given: Trevor
    - family: Mazumder
      given: Rahul
    - family: Lee
      given: Jason D.
    - family: Zadeh
      given: Reza
  citation-key: Hastie2015Matrix
  container-title: Journal of Machine Learning Research
  issued:
    - year: 2015
  page: 3367-3402
  title: Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares
  type: paper-conference
  URL: http://arxiv.org/abs/1410.2596
  volume: '16'

- id: Henderson1981Deriving
  accessed:
    - year: 2023
      month: 4
      day: 26
  author:
    - family: Henderson
      given: H. V.
    - family: Searle
      given: S. R.
  citation-key: Henderson1981Deriving
  container-title: SIAM Review
  container-title-short: SIAM Rev.
  DOI: 10.1137/1023004
  ISSN: 0036-1445
  issue: '1'
  issued:
    - year: 1981
      month: 1
  page: 53-60
  publisher: Society for Industrial and Applied Mathematics
  title: On Deriving the Inverse of a Sum of Matrices
  type: article-journal
  URL: >-
    https://ecommons.cornell.edu/bitstream/handle/1813/32749/BU-647-M.pdf;sequence=1
  volume: '23'

- id: Hoaglin1978Hat
  accessed:
    - year: 2023
      month: 4
      day: 21
  author:
    - family: Hoaglin
      given: David C.
    - family: Welsch
      given: Roy E.
  citation-key: Hoaglin1978Hat
  container-title: The American Statistician
  DOI: 10.2307/2683469
  ISSN: 0003-1305
  issue: '1'
  issued:
    - year: 1978
  page: 17-22
  publisher: '[American Statistical Association, Taylor & Francis, Ltd.]'
  title: The Hat Matrix in Regression and ANOVA
  type: article-journal
  URL: >-
    https://edisciplinas.usp.br/pluginfile.php/6189100/mod_resource/content/1/The%20Hat%20Matrix%20and%20ANOVA.pdf
  volume: '32'

- id: Kala1994Generalized
  accessed:
    - year: 2023
      month: 6
      day: 20
  author:
    - family: Kala
      given: Radoslaw
    - family: Klaczyński
      given: Krzysztof
  citation-key: Kala1994Generalized
  container-title: 'Sankhyā: The Indian Journal of Statistics, Series A (1961-2002)'
  ISSN: 0581-572X
  issue: '3'
  issued:
    - year: 1994
  page: 458-464
  publisher: Springer
  title: Generalized Inverses of a Sum of Matrices
  type: article-journal
  URL: https://www.jstor.org/stable/25051011
  volume: '56'

- id: Kannan2016Highperformance
  accessed:
    - year: 2016
      month: 10
      day: 14
  author:
    - family: Kannan
      given: Ramakrishnan
    - family: Ballard
      given: Grey
    - family: Park
      given: Haesun
  citation-key: Kannan2016Highperformance
  collection-title: PPoPP '16
  container-title: >-
    Proceedings of the 21st ACM SIGPLAN Symposium on Principles and Practice of
    Parallel Programming
  DOI: 10.1145/2851141.2851152
  event-place: New York, NY, USA
  ISBN: 978-1-4503-4092-2
  issued:
    - year: 2016
  page: 9:1–9:11
  publisher: ACM
  publisher-place: New York, NY, USA
  title: A High-performance Parallel Algorithm for Nonnegative Matrix Factorization
  type: paper-conference

- id: Kannan2016Scalable
  accessed:
    - year: 2016
      month: 10
      day: 14
  author:
    - family: Kannan
      given: Ramakrishnan
  citation-key: Kannan2016Scalable
  issued:
    - year: 2016
      month: 4
      day: 15
  title: Scalable and distributed constrained low rank approximations
  type: article-journal
  URL: https://smartech.gatech.edu/handle/1853/54962

- id: Kim2008Nonnegative
  accessed:
    - year: 2016
      month: 7
      day: 18
  author:
    - family: Kim
      given: H.
    - family: Park
      given: H.
  citation-key: Kim2008Nonnegative
  container-title: SIAM Journal on Matrix Analysis and Applications
  container-title-short: SIAM. J. Matrix Anal. & Appl.
  DOI: 10.1137/07069239X
  ISSN: 0895-4798
  issue: '2'
  issued:
    - year: 2008
      month: 1
      day: 1
  page: 713-730
  title: >-
    Nonnegative Matrix Factorization Based on Alternating Nonnegativity
    Constrained Least Squares and Active Set Method
  type: article-journal
  URL: http://www.cc.gatech.edu/~hpark/papers/simax-nmf.pdf
  volume: '30'

- id: Kumar2016Literature
  accessed:
    - year: 2016
      month: 7
      day: 4
  author:
    - family: Kumar
      given: N. Kishore
    - family: Shneider
      given: Jan
  citation-key: Kumar2016Literature
  container-title: arXiv:1606.06511 [cs, math]
  issued:
    - year: 2016
      month: 6
      day: 21
  title: Literature survey on low rank approximation of matrices
  type: article-journal
  URL: http://arxiv.org/abs/1606.06511

- id: Liberty2007Randomized
  author:
    - family: Liberty
      given: Edo
    - family: Woolfe
      given: Franco
    - family: Martinsson
      given: Per-Gunnar
    - family: Rokhlin
      given: Vladimir
    - family: Tygert
      given: Mark
  citation-key: Liberty2007Randomized
  container-title: Proceedings of the National Academy of Sciences
  container-title-short: PNAS
  DOI: 10.1073/pnas.0709640104
  ISSN: 0027-8424, 1091-6490
  issue: '51'
  issued:
    - year: 2007
      month: 12
      day: 18
  language: en
  page: 20167-20172
  PMID: '18056803'
  title: Randomized algorithms for the low-rank approximation of matrices
  type: article-journal
  volume: '104'

- id: Lim2007Variational
  author:
    - family: Lim
      given: Yew Jin
    - family: Teh
      given: Yee Whye
  citation-key: Lim2007Variational
  container-title: Proceedings of KDD cup and workshop
  issued:
    - year: 2007
  page: 15–21
  publisher: Citeseer
  title: Variational Bayesian approach to movie rating prediction
  type: paper-conference
  volume: '7'

- id: Lin2016Review
  accessed:
    - year: 2016
      month: 3
      day: 29
  author:
    - family: Lin
      given: Zhouchen
  citation-key: Lin2016Review
  issued:
    - year: 2016
  title: A Review on Low-Rank Models in Signal and Data Analysis
  type: article-journal
  URL: >-
    http://www.cis.pku.edu.cn/faculty/vision/zlin/Publications/2016-BDIA-LR_review.pdf

- id: Liu2015Performance
  author:
    - family: Liu
      given: T.
    - family: Tao
      given: D.
  citation-key: Liu2015Performance
  container-title: IEEE Transactions on Neural Networks and Learning Systems
  DOI: 10.1109/TNNLS.2015.2458986
  ISSN: 2162-237X
  issue: '99'
  issued:
    - year: 2015
  page: 1-1
  title: On the Performance of Manhattan Nonnegative Matrix Factorization
  type: article-journal
  volume: PP

- id: Lu2022Rigorous
  accessed:
    - year: 2023
      month: 4
      day: 21
  author:
    - family: Lu
      given: Jun
  citation-key: Lu2022Rigorous
  issued:
    - year: 2022
      month: 7
      day: 30
  language: en
  number: arXiv:2105.04240
  publisher: arXiv
  title: A rigorous introduction to linear models
  type: article
  URL: http://arxiv.org/abs/2105.04240

- id: Mahoney2010Randomized
  accessed:
    - year: 2016
      month: 6
      day: 20
  author:
    - family: Mahoney
      given: Michael W.
  citation-key: Mahoney2010Randomized
  DOI: 10.1561/2200000035
  issued:
    - year: 2010
  language: en
  title: Randomized Algorithms for Matrices and Data
  type: book
  URL: http://arxiv.org/abs/1104.5557
  volume: '3'

- id: Martinsson2016Randomized
  accessed:
    - year: 2016
      month: 8
      day: 4
  author:
    - family: Martinsson
      given: Per-Gunnar
  citation-key: Martinsson2016Randomized
  container-title: arXiv:1607.01649 [math]
  issued:
    - year: 2016
      month: 7
      day: 6
  title: >-
    Randomized methods for matrix computations and analysis of high dimensional
    data
  type: article-journal
  URL: http://arxiv.org/abs/1607.01649

- id: Miller1981Inverse
  accessed:
    - year: 2023
      month: 6
      day: 20
  author:
    - family: Miller
      given: Kenneth S.
  citation-key: Miller1981Inverse
  container-title: Mathematics Magazine
  DOI: 10.2307/2690437
  ISSN: 0025-570X
  issue: '2'
  issued:
    - year: 1981
  page: 67-72
  publisher: Mathematical Association of America
  title: On the Inverse of the Sum of Matrices
  type: article-journal
  volume: '54'

- id: Minka2000Old
  archive: Calibre
  author:
    - family: Minka
      given: Thomas P.
  citation-key: Minka2000Old
  issued:
    - year: 2000
  language: eng
  title: Old and new matrix algebra useful for statistics
  type: book
  URL: >-
    http://ce.sharif.edu/courses/93-94/1/ce717-2/resources/root/Review/minka-matrix.pdf

- id: Nakajima2012Theoretical
  author:
    - family: Nakajima
      given: Shinichi
    - family: Sugiyama
      given: Masashi
  citation-key: Nakajima2012Theoretical
  container-title: Journal of Machine Learning Research
  issued:
    - year: 2012
  language: en
  page: '66'
  title: Theoretical Analysis of Bayesian Matrix Factorization
  type: article-journal

- id: Nakatsukasa2019Lowrank
  accessed:
    - year: 2023
      month: 4
      day: 7
  author:
    - family: Nakatsukasa
      given: Yuji
  citation-key: Nakatsukasa2019Lowrank
  issued:
    - year: 2019
      month: 5
      day: 27
  language: en
  number: arXiv:1905.11490
  publisher: arXiv
  title: The low-rank eigenvalue problem
  type: article
  URL: http://arxiv.org/abs/1905.11490

- id: Nowak2013Kriging
  author:
    - family: Nowak
      given: W.
    - family: Litvinenko
      given: A.
  citation-key: Nowak2013Kriging
  container-title: Mathematical Geosciences
  container-title-short: Math Geosci
  DOI: 10.1007/s11004-013-9453-6
  ISSN: 1874-8961, 1874-8953
  issue: '4'
  issued:
    - year: 2013
      month: 5
      day: 1
  language: en
  page: 411-435
  title: >-
    Kriging and Spatial Design Accelerated by Orders of Magnitude: Combining
    Low-Rank Covariance Approximations with FFT-Techniques
  type: article-journal
  URL: >-
    https://www.researchgate.net/profile/Alexander_Litvinenko/publication/237332060_Kriging_and_spatial_design_accelerated_by_orders_of_magnitude_combining_low-rank_covariance_approximations_with_FFT-techniques/links/552163110cf2a2d9e1450a84.pdf
  volume: '45'

- id: Petersen2012Matrix
  accessed:
    - year: 2016
      month: 8
      day: 18
  author:
    - family: Petersen
      given: Kaare Brandt
    - family: Pedersen
      given: Michael Syskind
  citation-key: Petersen2012Matrix
  issued:
    - year: 2012
  title: The Matrix Cookbook
  type: report
  URL: http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=3274

- id: Rellich1954Perturbation
  accessed:
    - year: 2023
      month: 11
      day: 26
  author:
    - family: Rellich
      given: F.
  call-number: '000035194'
  citation-key: Rellich1954Perturbation
  contributor:
    - family: New York University
      given: Institute of Fine Arts Library
  issued:
    - year: 1954
  language: eng
  number-of-pages: '352'
  publisher: 'New York: Courant Institute of Mathematical Sciences, New York University'
  title: Perturbation theory of eigenvalue problems
  type: book
  URL: http://archive.org/details/perturbationtheo00rell

- id: Rokhlin2009Randomized
  author:
    - family: Rokhlin
      given: Vladimir
    - family: Szlam
      given: Arthur
    - family: Tygert
      given: Mark
  citation-key: Rokhlin2009Randomized
  container-title: SIAM J. Matrix Anal. Appl.
  DOI: 10.1137/080736417
  ISSN: 0895-4798
  issue: '3'
  issued:
    - year: 2009
      month: 8
  page: 1100–1124
  title: A Randomized Algorithm for Principal Component Analysis
  type: article-journal
  URL: http://tygert.com/blanczos.pdf
  volume: '31'

- id: Saad2003Iterative
  author:
    - family: Saad
      given: Yousef
  citation-key: Saad2003Iterative
  edition: 2nd
  ISBN: 978-0-89871-800-3
  issued:
    - year: 2003
      month: 1
      day: 1
  language: en
  number-of-pages: '538'
  publisher: SIAM
  title: 'Iterative Methods for Sparse Linear Systems: Second Edition'
  type: book
  URL: https://www-users.cse.umn.edu/~saad/IterMethBook_2ndEd.pdf

- id: Salakhutdinov2008Bayesian
  accessed:
    - year: 2017
      month: 12
      day: 5
  author:
    - family: Salakhutdinov
      given: Ruslan
    - family: Mnih
      given: Andriy
  citation-key: Salakhutdinov2008Bayesian
  collection-title: ICML '08
  container-title: Proceedings of the 25th International Conference on Machine Learning
  DOI: 10.1145/1390156.1390267
  event-place: New York, NY, USA
  ISBN: 978-1-60558-205-4
  issued:
    - year: 2008
  page: 880–887
  publisher: ACM
  publisher-place: New York, NY, USA
  title: Bayesian Probabilistic Matrix Factorization Using Markov Chain Monte Carlo
  type: paper-conference
  URL: http://www.machinelearning.org/archive/icml2008/papers/600.pdf

- id: Saul2023Geometrical
  accessed:
    - year: 2023
      month: 1
      day: 17
  author:
    - family: Saul
      given: Lawrence K.
  citation-key: Saul2023Geometrical
  container-title: Transactions on Machine Learning Research
  issued:
    - year: 2023
      month: 1
      day: 9
  language: en
  title: >-
    A geometrical connection between sparse and low-rank matrices and its
    application to manifold learning
  type: article-journal
  URL: https://openreview.net/forum?id=p8gncJbMit

- id: Seeger2004Low
  citation-key: Seeger2004Low
  editor:
    - family: Seeger
      given: Matthias
  issued:
    - year: 2004
  title: Low Rank Updates for the Cholesky Decomposition
  type: book

- id: Seshadhri2020Impossibility
  accessed:
    - year: 2020
      month: 5
      day: 2
  author:
    - family: Seshadhri
      given: C.
    - family: Sharma
      given: Aneesh
    - family: Stolman
      given: Andrew
    - family: Goel
      given: Ashish
  citation-key: Seshadhri2020Impossibility
  container-title: Proceedings of the National Academy of Sciences
  container-title-short: PNAS
  DOI: 10.1073/pnas.1911030117
  ISSN: 0027-8424, 1091-6490
  issue: '11'
  issued:
    - year: 2020
      month: 3
      day: 17
  language: en
  page: 5631-5637
  PMID: '32123073'
  publisher: National Academy of Sciences
  section: Physical Sciences
  title: >-
    The impossibility of low-rank representations for triangle-rich complex
    networks
  type: article-journal
  volume: '117'

- id: Shi2017Survey
  accessed:
    - year: 2022
      month: 3
      day: 29
  author:
    - family: Shi
      given: Jiarong
    - family: Zheng
      given: Xiuyun
    - family: Yang
      given: Wei
  citation-key: Shi2017Survey
  container-title: Entropy
  container-title-short: Entropy
  DOI: 10.3390/e19080424
  ISSN: 1099-4300
  issue: '8'
  issued:
    - year: 2017
      month: 8
      day: 19
  language: en
  page: '424'
  title: Survey on Probabilistic Models of Low-Rank Matrix Factorizations
  type: article-journal
  volume: '19'

- id: Srebro2004Maximummargin
  accessed:
    - year: 2022
      month: 2
      day: 3
  author:
    - family: Srebro
      given: Nathan
    - family: Rennie
      given: Jason D. M.
    - family: Jaakkola
      given: Tommi S.
  citation-key: Srebro2004Maximummargin
  collection-title: NIPS'04
  container-title: Advances in Neural Information Processing Systems
  event-place: Cambridge, MA, USA
  event-title: NIPS
  issued:
    - year: 2004
      month: 12
      day: 1
  page: 1329–1336
  publisher: MIT Press
  publisher-place: Cambridge, MA, USA
  title: Maximum-margin matrix factorization
  type: paper-conference
  URL: https://openreview.net/forum?id=Sk-3qvbu-S
  volume: '17'

- id: Sundin2016Bayesian
  author:
    - family: Sundin
      given: Martin
  citation-key: Sundin2016Bayesian
  genre: PhD Thesis
  issued:
    - year: 2016
  publisher: KTH Royal Institute of Technology
  title: Bayesian methods for sparse and low-rank matrix problems
  type: thesis

- id: Tropp2016Randomized
  accessed:
    - year: 2016
      month: 9
      day: 7
  author:
    - family: Tropp
      given: Joel A.
    - family: Yurtsever
      given: Alp
    - family: Udell
      given: Madeleine
    - family: Cevher
      given: Volkan
  citation-key: Tropp2016Randomized
  container-title: arXiv:1609.00048 [cs, math, stat]
  issued:
    - year: 2016
      month: 8
      day: 31
  title: Randomized single-view algorithms for low-rank matrix approximation
  type: article-journal
  URL: http://arxiv.org/abs/1609.00048

- id: Tropp2017Practical
  accessed:
    - year: 2018
      month: 8
      day: 10
  author:
    - family: Tropp
      given: Joel A.
    - family: Yurtsever
      given: Alp
    - family: Udell
      given: Madeleine
    - family: Cevher
      given: Volkan
  citation-key: Tropp2017Practical
  container-title: SIAM Journal on Matrix Analysis and Applications
  DOI: 10.1137/17M1111590
  ISSN: 0895-4798, 1095-7162
  issue: '4'
  issued:
    - year: 2017
      month: 1
  language: en
  page: 1454-1485
  title: Practical sketching algorithms for low-rank matrix approximation
  type: article-journal
  URL: http://arxiv.org/abs/1609.00048
  volume: '38'

- id: Tufts1982Estimation
  author:
    - family: Tufts
      given: D. W.
    - family: Kumaresan
      given: R.
  citation-key: Tufts1982Estimation
  container-title: Proceedings of the IEEE
  DOI: 10.1109/PROC.1982.12428
  ISSN: 0018-9219
  issue: '9'
  issued:
    - year: 1982
      month: 9
  page: 975-989
  title: >-
    Estimation of frequencies of multiple sinusoids: Making linear prediction
    perform like maximum likelihood
  type: article-journal
  volume: '70'

- id: Turkmen2015Review
  accessed:
    - year: 2015
      month: 8
      day: 6
  author:
    - family: Türkmen
      given: Ali Caner
  citation-key: Turkmen2015Review
  container-title: arXiv:1507.03194 [cs, stat]
  issued:
    - year: 2015
      month: 7
      day: 12
  title: A Review of Nonnegative Matrix Factorization Methods for Clustering
  type: article-journal
  URL: http://arxiv.org/abs/1507.03194

- id: Udell2019Why
  accessed:
    - year: 2019
      month: 5
      day: 21
  author:
    - family: Udell
      given: M.
    - family: Townsend
      given: A.
  citation-key: Udell2019Why
  container-title: SIAM Journal on Mathematics of Data Science
  container-title-short: SIAM Journal on Mathematics of Data Science
  DOI: 10.1137/18M1183480
  issue: '1'
  issued:
    - year: 2019
      month: 1
      day: 1
  page: 144-160
  title: Why Are Big Data Matrices Approximately Low Rank?
  type: article-journal
  volume: '1'

- id: Wilkinson2019EndEnd
  accessed:
    - year: 2019
      month: 2
      day: 3
  author:
    - family: Wilkinson
      given: William J.
    - family: Andersen
      given: Michael Riis
    - family: Reiss
      given: Joshua D.
    - family: Stowell
      given: Dan
    - family: Solin
      given: Arno
  citation-key: Wilkinson2019EndEnd
  container-title: arXiv:1901.11436 [cs, eess, stat]
  issued:
    - year: 2019
      month: 1
      day: 31
  language: en
  title: End-to-End Probabilistic Inference for Nonstationary Audio Analysis
  type: article-journal
  URL: https://arxiv.org/abs/1901.11436v1

- id: Woodruff2014Sketching
  archive: Calibre
  author:
    - family: Woodruff
      given: David P.
  citation-key: Woodruff2014Sketching
  collection-number: '1.0'
  collection-title: Foundations and Trends in Theoretical Computer Science
  DOI: 10.1561/0400000060
  ISBN: 978-1-68083-004-0
  issued:
    - year: 2014
      month: 11
      day: 14
  language: eng
  publisher: Now Publishers
  title: Sketching as a Tool for Numerical Linear Algebra
  type: book
  URL: http://researcher.watson.ibm.com/researcher/files/us-dpwoodru/wNow.pdf

- id: Woolfe2008Fast
  author:
    - family: Woolfe
      given: Franco
    - family: Liberty
      given: Edo
    - family: Rokhlin
      given: Vladimir
    - family: Tygert
      given: Mark
  citation-key: Woolfe2008Fast
  container-title: Applied and Computational Harmonic Analysis
  container-title-short: Applied and Computational Harmonic Analysis
  DOI: 10.1016/j.acha.2007.12.002
  ISSN: 1063-5203
  issue: '3'
  issued:
    - year: 2008
      month: 11
      day: 1
  page: 335-366
  title: A fast randomized algorithm for the approximation of matrices
  type: article-journal
  URL: http://www.tygert.com/frand.pdf
  volume: '25'

- id: XinghaoDing2011Bayesian
  accessed:
    - year: 2022
      month: 2
      day: 4
  author:
    - literal: Xinghao Ding
    - literal: Lihan He
    - family: Carin
      given: L.
  citation-key: XinghaoDing2011Bayesian
  container-title: IEEE Transactions on Image Processing
  container-title-short: IEEE Trans. on Image Process.
  DOI: 10.1109/TIP.2011.2156801
  ISSN: 1057-7149, 1941-0042
  issue: '12'
  issued:
    - year: 2011
      month: 12
  language: en
  page: 3419-3430
  title: Bayesian Robust Principal Component Analysis
  type: article-journal
  volume: '20'

- id: Yang2018Fast
  accessed:
    - year: 2022
      month: 7
      day: 26
  author:
    - family: Yang
      given: Linxiao
    - family: Fang
      given: Jun
    - family: Duan
      given: Huiping
    - family: Li
      given: Hongbin
    - family: Zeng
      given: Bing
  citation-key: Yang2018Fast
  container-title: IEEE Transactions on Signal Processing
  container-title-short: IEEE Trans. Signal Process.
  DOI: 10.1109/TSP.2018.2816575
  ISSN: 1053-587X, 1941-0476
  issue: '11'
  issued:
    - year: 2018
      month: 6
      day: 1
  language: en
  page: 2804-2817
  title: >-
    Fast Low-Rank Bayesian Matrix Completion with Hierarchical Gaussian Prior
    Models
  type: article-journal
  URL: http://arxiv.org/abs/1708.02455
  volume: '66'

- id: Yin2016Laplacian
  author:
    - family: Yin
      given: M.
    - family: Gao
      given: J.
    - family: Lin
      given: Z.
  citation-key: Yin2016Laplacian
  container-title: IEEE Transactions on Pattern Analysis and Machine Intelligence
  DOI: 10.1109/TPAMI.2015.2462360
  ISSN: 0162-8828
  issue: '3'
  issued:
    - year: 2016
      month: 3
  page: 504-517
  title: Laplacian Regularized Low-Rank Representation and Its Applications
  type: article-journal
  URL: >-
    http://www.cis.pku.edu.cn/faculty/vision/zlin/Publications/2015-TPAMI-LaplacianLRR.pdf
  volume: '38'

- id: Yu2012Scalable
  author:
    - family: Yu
      given: Hsiang-Fu
    - family: Hsieh
      given: Cho-Jui
    - family: Si
      given: Si
    - family: Dhillon
      given: Inderjit S.
  citation-key: Yu2012Scalable
  container-title: IEEE International Conference of Data Mining
  DOI: 10.1109/ICDM.2012.168
  event-title: 2012 IEEE 12th International Conference on Data Mining
  issued:
    - year: 2012
  page: 765-774
  title: >-
    Scalable Coordinate Descent Approaches to Parallel Matrix Factorization for
    Recommender Systems
  type: paper-conference
  URL: http://www.cs.utexas.edu/~rofuyu/papers/icdm-pmf.pdf

- id: Yu2014Parallel
  author:
    - family: Yu
      given: Hsiang-Fu
    - family: Hsieh
      given: Cho-Jui
    - family: Si
      given: Si
    - family: Dhillon
      given: Inderjit S.
  citation-key: Yu2014Parallel
  container-title: Knowledge and Information Systems
  container-title-short: Knowl Inf Syst
  DOI: 10.1007/s10115-013-0682-2
  ISSN: 0219-1377, 0219-3116
  issue: '3'
  issued:
    - year: 2014
      month: 12
      day: 1
  language: en
  page: 793-819
  title: Parallel matrix factorization for recommender systems
  type: article-journal
  URL: http://www.cs.utexas.edu/~rofuyu/papers/kais-pmf.pdf
  volume: '41'

- id: Yu2017$N
  author:
    - family: Yu
      given: Chenhan D.
    - family: March
      given: William B.
    - family: Biros
      given: George
  citation-key: Yu2017$N
  container-title: arXiv:1701.02324 [cs]
  event-title: 31st IEEE International Parallel & Distributed Processing Symposium
  issued:
    - year: 2017
      month: 1
      day: 9
  title: An $N \log N$ Parallel Fast Direct Solver for Kernel Matrices
  type: paper-conference
  URL: http://arxiv.org/abs/1701.02324

- id: Zhang2017Randomization
  accessed:
    - year: 2017
      month: 8
      day: 28
  author:
    - family: Zhang
      given: Kai
    - family: Liu
      given: Chuanren
    - family: Zhang
      given: Jie
    - family: Xiong
      given: Hui
    - family: Xing
      given: Eric
    - family: Ye
      given: Jieping
  citation-key: Zhang2017Randomization
  collection-title: KDD '17
  container-title: >-
    Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge
    Discovery and Data Mining
  DOI: 10.1145/3097983.3098050
  event-place: New York, NY, USA
  ISBN: 978-1-4503-4887-4
  issued:
    - year: 2017
  page: 615–623
  publisher: ACM
  publisher-place: New York, NY, USA
  title: >-
    Randomization or Condensation?: Linear-Cost Matrix Sketching Via Cascaded
    Compression Sampling
  type: paper-conference
  URL: http://doi.acm.org/10.1145/3097983.3098050

- id: Zhang2017Stochastic
  author:
    - family: Zhang
      given: Xiao
    - family: Wang
      given: Lingxiao
    - family: Gu
      given: Quanquan
  citation-key: Zhang2017Stochastic
  container-title: arXiv:1701.00481 [stat]
  issued:
    - year: 2017
      month: 1
      day: 2
  title: >-
    Stochastic Variance-reduced Gradient Descent for Low-rank Matrix Recovery
    from Linear Measurements
  type: article-journal
  URL: http://arxiv.org/abs/1701.00481

- id: Zhou2011GoDec
  accessed:
    - year: 2023
      month: 8
      day: 22
  author:
    - family: Zhou
      given: Tianyi
    - family: Tao
      given: Dacheng
  citation-key: Zhou2011GoDec
  collection-title: ICML'11
  container-title: >-
    Proceedings of the 28th International Conference on International Conference
    on Machine Learning
  event-place: Madison, WI, USA
  ISBN: 978-1-4503-0619-5
  issued:
    - year: 2011
      month: 6
      day: 28
  page: 33–40
  publisher: Omnipress
  publisher-place: Madison, WI, USA
  title: 'GoDec: randomized low-rank & sparse matrix decomposition in noisy case'
  type: paper-conference
  URL: http://www.icml-2011.org/papers/41_icmlpaper.pdf

- id: Zhou2012Multilabel
  accessed:
    - year: 2015
      month: 6
      day: 23
  author:
    - family: Zhou
      given: Tianyi
    - family: Tao
      given: Dacheng
  citation-key: Zhou2012Multilabel
  container-title: Journal of Machine Learning Research
  issued:
    - year: 2012
  title: Multi-label Subspace Ensemble
  type: article-journal
  URL: http://www.jmlr.org/proceedings/papers/v22/zhou12a/zhou12a.pdf
...
