Editing Principal component analysis (section)

=== Sparse PCA ===
{{main|Sparse PCA}}
A particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables. [[Sparse PCA]] overcomes this disadvantage by finding linear combinations that contain just a few input variables. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by adding sparsity constraint on the input variables.
Several approaches have been proposed, including
* a regression framework,<ref>
{{cite journal
 |author1=Hui Zou
 |author2=Trevor Hastie
 |author3=Robert Tibshirani
 |year=2006
 |title=Sparse principal component analysis
 |journal=[[Journal of Computational and Graphical Statistics]]
 |volume=15 |issue=2 |pages=262–286
 |url=http://www-stat.stanford.edu/~hastie/Papers/spc_jcgs.pdf
 |doi=10.1198/106186006x113430
|citeseerx=10.1.1.62.580
 |s2cid=5730904
 }}</ref>
* a convex relaxation/semidefinite programming framework,<ref name="SDP">
{{cite journal
 |author1=Alexandre d'Aspremont
 |author2=Laurent El Ghaoui
 |author3=Michael I. Jordan
 |author4=Gert R. G. Lanckriet
 |year=2007
 |title=A Direct Formulation for Sparse PCA Using Semidefinite Programming
 |journal=[[SIAM Review]]
 |volume=49 |issue=3 |pages=434–448
 |url=http://www.cmap.polytechnique.fr/~aspremon/PDF/sparsesvd.pdf
 |doi=10.1137/050645506
|arxiv=cs/0406021
 |s2cid=5490061
 }}</ref>
* a generalized power method framework<ref>
{{cite journal
 |author1=Michel Journee
 |author2=Yurii Nesterov
 |author3=Peter Richtarik
 |author4=Rodolphe Sepulchre
 |year=2010
 |title=Generalized Power Method for Sparse Principal Component Analysis
 |journal=[[Journal of Machine Learning Research]]
 |volume=11 |pages=517–553
 |url=http://jmlr.csail.mit.edu/papers/volume11/journee10a/journee10a.pdf
 |arxiv=0811.4724
 |id=CORE Discussion Paper 2008/70
|bibcode=2008arXiv0811.4724J
 }}</ref>
* an alternating maximization framework<ref>
{{cite arXiv
 |author1=Peter Richtarik
 |author2=Martin Takac
 |author3=S. Damla Ahipasaoglu
 |year=2012
 |title=Alternating Maximization: Unifying Framework for 8 Sparse PCA Formulations and Efficient Parallel Codes
 |eprint=1212.4137
 |class=stat.ML
}}</ref>
* forward-backward greedy search and exact methods using branch-and-bound techniques,<ref>
{{cite conference
 |author1=Baback Moghaddam
 |author2=Yair Weiss
 |author3=Shai Avidan
 |year=2005
 |chapter=Spectral Bounds for Sparse PCA: Exact and Greedy Algorithms
 |title=Advances in Neural Information Processing Systems
 |volume=18
 |publisher=MIT Press
 |chapter-url=http://books.nips.cc/papers/files/nips18/NIPS2005_0643.pdf
}}</ref>
* Bayesian formulation framework.<ref>
{{cite journal
 |author1=Yue Guan
 |author2=Jennifer Dy|author2-link=Jennifer Dy
 |year=2009
 |title=Sparse Probabilistic Principal Component Analysis
 |journal=[[Journal of Machine Learning Research Workshop and Conference Proceedings]]
 |volume=5 |page=185
 |url=http://jmlr.csail.mit.edu/proceedings/papers/v5/guan09a/guan09a.pdf
}}</ref>
The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies were recently reviewed in a survey paper.<ref>
{{cite journal
 |author1=Hui Zou
 |author2=Lingzhou Xue
 |year=2018
 |title=A Selective Overview of Sparse Principal Component Analysis
 |journal=[[Proceedings of the IEEE]]
 |volume=106 |issue=8
 |pages=1311–1320
 |doi=10.1109/JPROC.2018.2846588
 |doi-access=free
 }}</ref>