Treelets--An adaptive multi-scale basis for sparse unordered data

TL;DR

Treelets are an adaptive multi-scale basis method designed for high-dimensional, noisy, and unordered data, enabling effective dimensionality reduction and feature selection by capturing the data's internal hierarchical structure.

Contribution

The paper introduces treelets, a novel adaptive hierarchical basis construction that extends wavelets to nonsmooth signals and is tailored for sparse, unordered high-dimensional data.

Findings

Treelets outperform PCA in various data scenarios.

Treelets effectively identify variable groupings and structures.

The method is simple to implement and has solid theoretical foundations.

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.