Sequential Learning of Principal Curves: Summarizing Data Streams on the Fly

Numerous methods have been proposed in the statistics and machine learning literature to sum up information and represent data by condensed and simpler to understand quantities. Among those methods, Principal Component Analysis (PCA) aims at identifying the maximal variance axes of data. This serves as a way to represent data in a more compact fashion and hopefully reveal as well as possible their variability. PCA has been introduced by Pearson (1901) and Spearman (1904) and further developed by Hotelling (1933). This is one of the most widely used procedures in multivariate exploratory analysis targeting dimension reduction or features extraction. Nonetheless, PCA is a linear procedure and the need for more sophisticated nonlinear techniques has led to the notion of principal curve. Principal curves may be seen as a nonlinear generalization of the first principal component. The goal is to obtain a curve which passes "in the middle" of data, as illustrated by Figure 1. This notion has been at the heart of numerous applications in many different domains, such as physics (Brunsdon, 2007; Friedsam and Oren, 1989), character and speech recognition (Kégl and Krzyżak, 2002; Reinhard and Niranjan, 1999), mapping and geology (Banfield and Raftery, 1992; Brunsdon, 2007; Stanford and Raftery, 2000), to name but a few.