# Sparse PCA

Sparse PCA (sparse principal component analysis) is a specialised technique used in statistical analysis and, in particular, in the analysis of multivariate data sets.

Ordinary principal component analysis (PCA) uses a vector space transform used to reduce multidimensional data sets to lower dimensions for analysis. It finds linear combinations of variables (called "principal components") that correspond to directions of maximal variance in the data. The number of new variables created by these linear combinations is usually much lower than the number of variables in the original dataset. Sparse PCA finds sets of sparse vectors for use as weights in the linear combinations while still explaining most of the variance present in the data.

Several approaches have been proposed, including a regression framework,[1] a convex relaxation/semidefinite programming framework,[2] a generalized power method framework[3] an alternating maximization framework[4] forward/backward greedy search and exact methods using branch-and-bound techniques,[5] Bayesian formulation framework.[6]

## References

1. ^ H. Zou and T. Hastie and R. Tibshirani (2006). "Sparse principal component analysis". Jcgs 2006 15(2): 262-286.
2. ^ Alexandre d’Aspremont, Laurent El Ghaoui, Michael I. Jordan, Gert R. G. Lanckriet (2004). "A Direct Formulation for Sparse PCA Using Semidefinite Programming". Advances in Neural Information Processing Systems (NIPS), MIT Press.
3. ^ Michel Journee, Yurii Nesterov, Peter Richtarik, Rodolphe Sepulchre (2008). "Generalized Power Method for Sparse Principal Component Analysis". CORE Discussion Paper 2008/70, Journal of Machine Learning Research 11 (2010) 517-553 0811: 4724. arXiv:0811.4724.
4. ^ Peter Richtarik, Martin Takac and S. Damla Ahipasaoglu (2012). Alternating Maximization: Unifying Framework for 8 Sparse PCA Formulations and Efficient Parallel Codes. arXiv:1212.4137.
5. ^ Baback Moghaddam, Yair Weiss, Shai Avidan (2005). "Spectral Bounds for Sparse PCA: Exact and Greedy Algorithms". Advances in Neural Information Processing Systems (NIPS), MIT Press.
6. ^ Yue Guan, Jennifer Dy (2009). "Sparse Probabilistic Principal Component Analysis". Journal of Machine Learning Research Workshop and Conference Proceedings. 5: AISTATS 2009.