## Abstract

We introduce a framework for representing functions defined on high-dimensional data. In this framework, we propose to use the eigenvectors of the graph Laplacian to construct a multiresolution analysis on the data. We assume the dataset to have an associated hierarchical tree partition, together with a function that measures the similarity between pairs of points in the dataset. The construction results in a one parameter family of orthonormal bases, which includes both the Haar basis as well as the eigenvectors of the graph Laplacian, as its two extremes. We describe a fast discrete transform for the expansion in any of the bases in this family, and estimate the decay rate of the expansion coefficients. We also bound the error of non-linear approximation of functions in our bases. The properties of our construction are demonstrated using various numerical examples.

Original language | English |
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Pages (from-to) | 420-451 |

Number of pages | 32 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 38 |

Issue number | 3 |

DOIs | |

State | Published - 1 May 2015 |

## Keywords

- Graph Laplacian
- High dimensional data
- Multiresolution analysis
- Multiwavelets