A Pure L1-norm Principal Component Analysis

Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284.
Computational Statistics & Data Analysis (Impact Factor: 1.4). 05/2013; 61:83-98. DOI: 10.1016/j.csda.2012.11.007
Source: PubMed


The L 1 norm has been applied in numerous variations of principal component analysis (PCA). L 1-norm PCA is an attractive alternative to traditional L 2-based PCA because it can impart robustness in the presence of outliers and is indicated for models where standard Gaussian assumptions about the noise may not apply. Of all the previously-proposed PCA schemes that recast PCA as an optimization problem involving the L 1 norm, none provide globally optimal solutions in polynomial time. This paper proposes an L 1-norm PCA procedure based on the efficient calculation of the optimal solution of the L 1-norm best-fit hyperplane problem. We present a procedure called L 1-PCA* based on the application of this idea that fits data to subspaces of successively smaller dimension. The procedure is implemented and tested on a diverse problem suite. Our tests show that L 1-PCA* is the indicated procedure in the presence of unbalanced outlier contamination.

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Available from: Edward L. Boone, Apr 10, 2014
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    • "One can apply standard non-linear optimization schemes to (1.4), e.g., sequential rank-one updates [19], alternating optimization [20] (a.k.a. coordinate descent), the Wiberg algorithm [10], augmented Lagrangian approaches [36], successive projections on hyperplanes and linear programming [5], to cite a few. The main drawback of this class of methods is that it does not guarantee to recover the global optimum of (1.4) and is in general sensitive to initialization. "
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    • "A line of recent research pursues calculation of L 1 principal components under error minimization [3]-[9]. The error surface is non-smooth and the problem non-convex resisting attempts to guaranteed optimization even with exponential computational cost. "
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