Some theoretical results on nonlinear principal component analysis (1998) [9 citations — 0 self]
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@TECHREPORT{Malthouse98sometheoretical,author = {Edward C. Malthouse},
title = {Some theoretical results on nonlinear principal component analysis},
institution = {In Proceedings of the American Control Conference},
year = {1998}
}
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Abstract:
Nonlinear principal components analysis (NLPCA) neural networks are feedforward autoassociative networks with five layers. The third layer has fewer nodes than the input or output layers. NLPCA has been shown to give better solutions to several feature extraction problems than existing methods, but very little is know about the theoretical properties of this method or its estimates. This paper studies NLPCA. It proposes a geometric interpretation by showing that NLPCA fits a lower-dimensional curve or surface through the training data. The first three layers project observations onto the curve or surface giving scores. The last three layers define the curve or surface. The first three layers are a continuous function, which I show has several implications: NLPCA "projections " are suboptimal producing larger approximation error, NLPCA is unable to model curves and surfaces that intersect themselves, and NLPCA cannot parameterize curves with parameterizations having discontinuous jumps. I establish results on the identification of score values and discuss their implications on interpreting score values. I discuss the relationship between NLPCA and principal curves and surfaces, another nonlinear feature extraction method.
