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Ridgelets: estimating with ridge functions

Candès, Emmanuel J. (2003) Ridgelets: estimating with ridge functions. Annals of Statistics, 31 (5). pp. 1561-1599. ISSN 0090-5364. http://resolver.caltech.edu/CaltechAUTHORS:CANaos03

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Abstract

Feedforward neural networks, projection pursuit regression, and more generally, estimation via ridge functions have been proposed as an approach to bypass the curse of dimensionality and are now becoming widely applied to approximation or prediction in applied sciences. To address problems inherent to these methods--ranging from the construction of neural networks to their efficiency and capability -- Candès [Appl. Comput. armon. Anal. 6 (1999) 197-218] developed a new system that allows the representation of arbitrary functions as superpositions of specific ridge functions, the ridgelets. In a nonparametric regression setting, this article suggests expanding noisy data into a ridgelet series and applying a scalar nonlinearity to the coefficients (damping); this is unlike existing approaches based on stepwise additions of elements. The procedure is simple, constructive, stable and spatially adaptive -- and fast algorithms have been developed to implement it. The ridgelet estimator is nearly optimal for estimating functions with certain kinds of spatial inhomogeneities. In addition, ridgelets help to identify new classes of estimands--corresponding to a new notion of smoothness--that are well suited for ridge functions estimation. While the results are stated in a decision theoretic framework, numerical experiments are also presented to illustrate the practical performance of the methodology.


Item Type:Article
Additional Information:2003 © Institute of Mathematical Statistics. Received October 1999; revised November 2002. Supported by NSF Grant DMS-95-05151 and by AFOSR MURI 95-P49620-96-1-0028. I am especially grateful to David Donoho for serving as my adviser and for many fruitful discussions. It is a pleasure to acknowledge conversations with Iain Johnstone. I would like to thank the anonymous referees for their role in improving the original manuscript.
Subject Keywords:Nonparametric regression, ridgelets, ridge functions, projection pursuit regression, minimax decision theory, radon transform, spatial inhomogeneities, edges, thresholding of ridgelet coefficients
Record Number:CaltechAUTHORS:CANaos03
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:CANaos03
Alternative URL:http://dx.doi.org/10.1214/aos/1065705119
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9057
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:24 Oct 2007
Last Modified:26 Dec 2012 09:45

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