Sequential Principal Curves Analysis

This report includes all the technical details of the Sequential Principal Curves Analysis (SPCA) in a single document. SPCA is an unsupervised nonlinear and invertible feature extraction technique. The identified curvilinear features can be interpreted as a set of nonlinear sensors: the response of each sensor is the projection onto the corresponding feature. Moreover, it can be easily tuned for different optimization criteria –e.g. infomax, error minimization, decorrelation– by choosing the right way to measure distances along each curvilinear feature. Even though proposed in [1] and shown to work in multiple modalities in [2], the SPCA framework has its original roots in the nonlinear ICA algorithm in [3]. Later on, the SPCA philosophy for nonlinear generalization of PCA originated substantially faster alternatives at the cost of introducing different constraints in the model. Namely, the Principal Polynomial Analysis (PPA) [4], and the Dimensionality Reduction via Regression (DRR) [5]. This report illustrates the reasons why we developed such family and is the appropriate technical companion for the missing details in [1], [2]. See also the data, code and examples in the dedicated sites http://isp.uv.es/spca.html and http://isp.uv.es/after effects.html