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Linear feature extraction at the presence of nonlinear dependencies among thedata is a fundamental challenge in unsupervised learning. We propose using aProbabilistic Gram-Schmidt (PGS) type orthogonalization process in order todetect and map out redundant dimensions. Specifically, by applying the PGSprocess over any family of functions which presumably captures the nonlineardependencies in the data, we construct a series of covariance matrices that caneither be used to remove those dependencies from the principal components, orto identify new large-variance directions. In the former case, we prove thatunder certain assumptions the resulting algorithms detect and remove nonlineardependencies whenever those dependencies lie in the linear span of the chosenfunction family. In the latter, we provide information-theoretic guarantees interms of entropy reduction. Both proposed methods extract linear features fromthe data while removing nonlinear redundancies. We provide simulation resultson synthetic and real-world datasets which show improved performance over PCAand state-of-the-art linear feature extraction algorithms, both in terms ofvariance maximization of the extracted features, and in terms of improvedperformance of classification algorithms.

Beyond PCA: A Probabilistic Gram-Schmidt Approach to Feature Extraction