Principal Component Analysis and Optimization: A Tutorial

@inproceedings{Reris2015PrincipalCA,
  title={Principal Component Analysis and Optimization: A Tutorial},
  author={Robert Reris and J. Paul Brooks},
  year={2015}
}
Principal component analysis (PCA) is one of the most widely used multivariate tech- niques in statistics. It is commonly used to reduce the dimensionality of data in order to examine its underlying structure and the covariance/correlation structure of a set of variables. While singular value decomposition provides a simple means for identi- cation of the principal components (PCs) for classical PCA, solutions achieved in this manner may not possess certain desirable properties including… 

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References

SHOWING 1-10 OF 34 REFERENCES

K-means clustering via principal component analysis

It is proved that principal components are the continuous solutions to the discrete cluster membership indicators for K-means clustering, which indicates that unsupervised dimension reduction is closely related to unsuper supervised learning.

A Pure L1-norm Principal Component Analysis.

Tests show that L1-PCA* is the indicated procedure in the presence of unbalanced outlier contamination and the application of this idea that fits data to subspaces of successively smaller dimension is presented.

Principal Component Analysis

  • I. Jolliffe
  • Mathematics, Geology
    International Encyclopedia of Statistical Science
  • 1986
Introduction * Properties of Population Principal Components * Properties of Sample Principal Components * Interpreting Principal Components: Examples * Graphical Representation of Data Using

Robust Principal Component Analysis with Non-Greedy l1-Norm Maximization

Experimental results on real world datasets show that the nongreedy method always obtains much better solution than that of the greedy method, and then a robust principal component analysis with non-greedy l1-norm maximization is proposed.

A pure L1L1-norm principal component analysis

Robust principal component analysis?

It is proved that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, this suggests the possibility of a principled approach to robust principal component analysis.

A Generalized Least-Square Matrix Decomposition

By finding the best low-rank approximation of the data with respect to a transposable quadratic norm, the generalized least-square matrix decomposition (GMD), directly accounts for structural relationships and is demonstrated for dimension reduction, signal recovery, and feature selection with high-dimensional structured data.

Principal Component Analysis Based on L1-Norm Maximization

  • Nojun Kwak
  • Computer Science
    IEEE Transactions on Pattern Analysis and Machine Intelligence
  • 2008
A method of principal component analysis (PCA) based on a new L1-norm optimization technique which is robust to outliers and invariant to rotations and also proven to find a locally maximal solution.

Spectral Relaxation for K-means Clustering

It is shown that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by Computing a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by computing a pivoted QR decomposition ofThe eigenvector matrix.

Principal Component Analysis

  • H. Shen
  • Environmental Science
    Encyclopedia of Database Systems
  • 2009
The Karhunen-Lo eve basis functions, more frequently referred to as principal components or empirical orthogonal functions (EOFs), of the noise response of the climate system are an important tool