کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4605057 | 1337542 | 2015 | 32 صفحه PDF | دانلود رایگان |
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.
Journal: Applied and Computational Harmonic Analysis - Volume 38, Issue 3, May 2015, Pages 420–451