**ABSTRACT**
### Hierarchical Parallel Matching Pursuit

#### Hans G. Feichtinger, Andreas Türk and Thomas Strohmer

The by now well-known matching pursuit method of S. Mallat
as well as the recently proposed orthogonal matching pursuit
work purely sequential and are based on the idea of choosing only one
single
atom at a given time. Pursuing ideas which are related to modifications of
the POCS method we suggest
a new type of orthogonalization procedure which allows to operate in
parallel
at different "levels". More precisely, we assume that we have a
dictionary
which consists of similar "pages", i.e. those pages are collection of
functions, generated from a single function by translations along a
subgroup which is the same for all such pages. Based on a certain
hierarchical structure (preference of pages) we apply an appropriate
Gram-Schmidt type orthogonalization procedure which allows to deal
with the matching pursuit problem in parallel at different levels.
After carrying out appropriate approximations at the different,
now orthogonal levels, one comes back to a representation based on
the given family of atoms, in a straight-forward way.

**Keywords:** algorithms, wavelets

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