Abstract
Reconstructing bandlimited functions from random sampling is an important problem in signal processing. Strohmer and Vershynin obtained good results for this problem by using a randomized version of the Kaczmarz algorithm (RK) and assigning to every equation a probability weight proportional to the average distance of the sample from its two nearest neighbors. However, their results are valid only for moderate to high sampling rates; in practice, it may not always be possible to obtain many samples. Experiments show that the number of projections required by RK and other Kaczmarz variants rises seemingly exponentially when the equations/variables ratio (EVR) falls below 5. CGMN, which is a CG acceleration of Kaczmarz, provides very good results for low values of EVR and it is much better than CGNR and CGNE. A derandomization method, based on an extension of the bit-reversal permutation, is combined with the weights and shown to improve the performance of CGMN and the regular (cyclic) Kaczmarz, which even outperforms RK. A byproduct of our results is the finding that signals composed mainly of high-frequency components are easier to recover.
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The author would like to thank the anonymous reviewers for their helpful comments. Section 4 was added in response to the issues raised by one of the reviewers.
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Gordon, D. A derandomization approach to recovering bandlimited signals across a wide range of random sampling rates. Numer Algor 77, 1141–1157 (2018). https://doi.org/10.1007/s11075-017-0356-3
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DOI: https://doi.org/10.1007/s11075-017-0356-3