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June 2008 Treelets—An adaptive multi-scale basis for sparse unordered data
Ann B. Lee, Boaz Nadler, Larry Wasserman
Ann. Appl. Stat. 2(2): 435-471 (June 2008). DOI: 10.1214/07-AOAS137

Abstract

In many modern applications, including analysis of gene expression and text documents, the data are noisy, high-dimensional, and unordered—with no particular meaning to the given order of the variables. Yet, successful learning is often possible due to sparsity: the fact that the data are typically redundant with underlying structures that can be represented by only a few features. In this paper we present treelets—a novel construction of multi-scale bases that extends wavelets to nonsmooth signals. The method is fully adaptive, as it returns a hierarchical tree and an orthonormal basis which both reflect the internal structure of the data. Treelets are especially well-suited as a dimensionality reduction and feature selection tool prior to regression and classification, in situations where sample sizes are small and the data are sparse with unknown groupings of correlated or collinear variables. The method is also simple to implement and analyze theoretically. Here we describe a variety of situations where treelets perform better than principal component analysis, as well as some common variable selection and cluster averaging schemes. We illustrate treelets on a blocked covariance model and on several data sets (hyperspectral image data, DNA microarray data, and internet advertisements) with highly complex dependencies between variables.

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Ann B. Lee. Boaz Nadler. Larry Wasserman. "Treelets—An adaptive multi-scale basis for sparse unordered data." Ann. Appl. Stat. 2 (2) 435 - 471, June 2008. https://doi.org/10.1214/07-AOAS137

Information

Published: June 2008
First available in Project Euclid: 3 July 2008

zbMATH: 05591278
MathSciNet: MR2524336
Digital Object Identifier: 10.1214/07-AOAS137

Keywords: dimensionality reduction , Feature selection , hierarchical clustering , local best basis , multi-resolution analysis , Principal Component Analysis , small sample sizes , Sparsity

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.2 • No. 2 • June 2008
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