Multiscale data sampling and function extension

Amit Bermanis, Amir Averbuch*, Ronald R. Coifman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We introduce a multiscale scheme for sampling scattered data and extending functions defined on the sampled data points, which overcomes some limitations of the Nyström interpolation method. The multiscale extension (MSE) method is based on mutual distances between data points. It uses a coarse-to-fine hierarchy of the multiscale decomposition of a Gaussian kernel. It generates a sequence of subsamples, which we refer to as adaptive grids, and a sequence of approximations to a given empirical function on the data, as well as their extensions to any newly-arrived data point. The subsampling is done by a special decomposition of the associated Gaussian kernel matrix in each scale in the hierarchical procedure.

Original languageEnglish
Pages (from-to)15-29
Number of pages15
JournalApplied and Computational Harmonic Analysis
Issue number1
StatePublished - Jan 2013


  • Diffusion maps
  • Gaussian kernel
  • Geometric harmonics
  • Multiscale
  • Nyström extension
  • Subsampling


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