Coding-Theoretic Methods for Sparse Recovery

Coding-Theoretic Methods for Sparse Recovery. Cheraghchi, M. In Proceedings of the 49th Annual Allerton Conference on Communication, Control, and Computing, pages 909–916, 2011.

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We review connections between coding-theoretic objects and sparse learning problems. In particular, we show how seemingly different combinatorial objects such as error-correcting codes, combinatorial designs, spherical codes, compressed sensing matrices and group testing designs can be obtained from one another. The reductions enable one to translate upper and lower bounds on the parameters attainable by one object to another. We survey some of the well-known reductions in a unified presentation, and bring some existing gaps to attention. New reductions are also introduced; in particular, we bring up the notion of minimum "L-wise distance" of codes and show that this notion closely captures the combinatorial structure of RIP-2 matrices. Moreover, we show how this weaker variation of the minimum distance is related to combinatorial list-decoding properties of codes.

@INPROCEEDINGS{ref:conf:Che11,
  author =	 {Mahdi Cheraghchi},
  title =	 {Coding-Theoretic Methods for Sparse Recovery},
  year =	 2011,
  booktitle =	 {Proceedings of the 49th {Annual Allerton Conference
                  on Communication, Control, and Computing}},
  pages =	 {909--916},
  doi =		 {10.1109/Allerton.2011.6120263},
  url_Link =	 {https://ieeexplore.ieee.org/document/6120263},
  url_Paper =	 {https://arxiv.org/abs/1110.0279},
  abstract =	 { We review connections between coding-theoretic
                  objects and sparse learning problems. In particular,
                  we show how seemingly different combinatorial
                  objects such as error-correcting codes,
                  combinatorial designs, spherical codes, compressed
                  sensing matrices and group testing designs can be
                  obtained from one another. The reductions enable one
                  to translate upper and lower bounds on the
                  parameters attainable by one object to another. We
                  survey some of the well-known reductions in a
                  unified presentation, and bring some existing gaps
                  to attention. New reductions are also introduced; in
                  particular, we bring up the notion of minimum
                  "L-wise distance" of codes and show that this notion
                  closely captures the combinatorial structure of
                  RIP-2 matrices. Moreover, we show how this weaker
                  variation of the minimum distance is related to
                  combinatorial list-decoding properties of codes.  }
}

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