Abstract
When confronted with massive data streams, summarizing data with dimension reduction methods such as PCA raises theoretical and algorithmic pitfalls. A principal curve acts as a nonlinear generalization of PCA, and the present paper proposes a novel algorithm to automatically and sequentially learn principal curves from data streams. We show that our procedure is supported by regret bounds with optimal sublinear remainder terms. A greedy local search implementation (called slpc, for sequential learning principal curves) that incorporates both sleeping experts and multi-armed bandit ingredients is presented, along with its regret computation and performance on synthetic and real-life data.
Highlights
Learning of Principal Curves: 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
principal component analysis (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
Summary
Learning of Principal Curves: 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. The goal is to obtain a curve which passes “in the middle” of data, as illustrated by Figure 1 This notion of skeletonization of data clouds has been at the heart of numerous applications in many different domains, such as physics [4,5], character and speech recognition [6,7], mapping and geology [5,8,9], to name but a few.
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