L1-norm principal component analysis
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L1-norm principal component analysis (L1-PCA) is a optimization problem and method for multivariate data analysis. L1-PCA is often preferred over standard principal component analysis (PCA), when the analyzed data may contain outliers (faulty points, irregular corruptions).
In principle, PCA seeks a collection of orthogonal directions that define a subspace wherein data representation is maximized. Standard PCA quantifies data representation as the aggregate L2-norm of the projections of the data points into the subspace, or equivalently the aggregate Euclidean distance of the original points from their subspace-projected representations. Thefeore, standard PCA is also referred to L2-PCA, mostly to be distinguished from L1-PCA. Among the advantages of PCA that have contributed to its high popularity are its low-cost implementation by means of singular-value decomposition (SVD) and its quality approximation of the maximum-variance subspace of the data source, for certain data distributions, such as multivariate Gaussian, when it operates on sufficiently many nominal data points.
However, in modern big data sets, data often include grossly corrupted, faulty points, commonly referred to as outliers. Regretfully, standard PCA is known to be very sensitive against outliers, even when they appear as a small fraction of the processed data. The reason is the L2 formulation of PCA places squared emphasis to the magnitude of each coordinate of each data point, ultimately benefiting peripheral points, such as outliers. On the other hand, following an L1-norm formulation, L1-PCA places linear emphasis on the coordinates of each data point, thus counteracting and effectively restraining outliers.