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A nonlinear PCA based on manifold approximation

Abstract : We address the problem of generalizing Principal Component Analysis (PCA) from the approximation point of view. Given a data set in a high dimensional space, PCA proposes approximations by linear subspaces. These linear models can show some limits when the data distribution is not Gaussian.To overcome these limits, we present Auto-Associative Composite (AAC) models based on manifold approximation. AAC models benefit from interesting theoretical properties, generalizing PCA ones. We take profit of these properties to propose an iterative algorithm to compute the manifold, and prove its convergence in a finite number of steps. PCA models and AAC models are first compared on a theoretical point of view. As a result, we show that PCA is the unique additive AAC model. Then a practical comparison of AAC and PCA models is presented on a data set made of curves.
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https://hal.inria.fr/hal-00724764
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Submitted on : Wednesday, August 22, 2012 - 3:28:58 PM
Last modification on : Thursday, January 20, 2022 - 5:30:12 PM
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  • HAL Id : hal-00724764, version 1

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Stéphane Girard. A nonlinear PCA based on manifold approximation. Computational Statistics, 2000, 15 (2), pp.145-167. ⟨hal-00724764⟩

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