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Iterative thresholding for sparse approximations. (English) Zbl 1175.94060
Summary: Sparse signal expansions represent or approximate a signal using a small number of elements from a large collection of elementary waveforms. Finding the optimal sparse expansion is known to be NP hard in general and non-optimal strategies such as matching pursuit, orthogonal matching pursuit, basis pursuit and basis pursuit de-noising are often called upon. These methods show good performance in practical situations, however, they do not operate on the $$\ell _{0}$$ penalised cost functions that are often at the heart of the problem.
In this paper we study two iterative algorithms that are minimising the cost functions of interest. Furthermore, each iteration of these strategies has computational complexity similar to a matching pursuit iteration, making the methods applicable to many real world problems. However, the optimisation problem is non-convex and the strategies are only guaranteed to find local solutions, so good initialisation becomes paramount. We here study two approaches. The first approach uses the proposed algorithms to refine the solutions found with other methods, replacing the typically used conjugate gradient solver. The second strategy adapts the algorithms and we show on one example that this adaptation can be used to achieve results that lie between those obtained with matching pursuit and those found with orthogonal matching pursuit, while retaining the computational complexity of the matching pursuit algorithm.

##### MSC:
 94A20 Sampling theory in information and communication theory 94A13 Detection theory in information and communication theory 68Q25 Analysis of algorithms and problem complexity 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 90C27 Combinatorial optimization 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 68W25 Approximation algorithms 68W40 Analysis of algorithms
PDCO; DT-CWT
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