Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes

Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes. Cheraghchi, M., Guruswami, V., & Velingker, A. SIAM Journal on Computing, 42(5):1888–1914, 2013. Preliminary version in Proceedings of SODA 2013.

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We prove that a random linear code over $\mathbb{F}_q$, with probability arbitrarily close to $1$, is list decodable at radius $1-1/q-ε$ with list size $L=O(1/ε^2)$ and rate $R=Ω_q(ε^2/(łog^3(1/ε)))$. Up to the polylogarithmic factor in $1/ε$ and constant factors depending on $q$, this matches the lower bound $L=Ω_q(1/ε^2)$ for the list size and upper bound $R=O_q(ε^2)$ for the rate. Previously only existence (and not abundance) of such codes was known for the special case $q=2$ (Guruswami, Håstad, Sudan and Zuckerman, 2002). In order to obtain our result, we employ a relaxed version of the well known Johnson bound on list decoding that translates the average Hamming distance between codewords to list decoding guarantees. We furthermore prove that the desired average-distance guarantees hold for a code provided that a natural complex matrix encoding the codewords satisfies the Restricted Isometry Property with respect to the Euclidean norm (RIP-2). For the case of random binary linear codes, this matrix coincides with a random submatrix of the Hadamard-Walsh transform matrix that is well studied in the compressed sensing literature. Finally, we improve the analysis of Rudelson and Vershynin (2008) on the number of random frequency samples required for exact reconstruction of $k$-sparse signals of length $N$. Specifically, we improve the number of samples from $O(k łog(N) łog^2(k) (łog k + łogłog N))$ to $O(k łog(N) · łog^3(k))$. The proof involves bounding the expected supremum of a related Gaussian process by using an improved analysis of the metric defined by the process. This improvement is crucial for our application in list decoding.

@ARTICLE{ref:CGV13,
  author =	 {Cheraghchi, Mahdi and Guruswami, Venkatesan and
                  Velingker, Ameya},
  title =	 {Restricted Isometry of {Fourier} Matrices and List
                  Decodability of Random Linear Codes},
  journal =	 {SIAM Journal on Computing},
  volume =	 42,
  number =	 5,
  pages =	 {1888--1914},
  year =	 2013,
  doi =		 {10.1137/120896773},
  keywords =	 {combinatorial list decoding, compressed sensing,
                  {Gaussian} processes},
  url_Link =	 {https://epubs.siam.org/doi/abs/10.1137/120896773},
  abstract =	 {We prove that a random linear code over
                  $\mathbb{F}_q$, with probability arbitrarily close
                  to $1$, is list decodable at radius $1-1/q-\epsilon$
                  with list size $L=O(1/\epsilon^2)$ and rate
                  $R=\Omega_q(\epsilon^2/(\log^3(1/\epsilon)))$. Up to
                  the polylogarithmic factor in $1/\epsilon$ and
                  constant factors depending on $q$, this matches the
                  lower bound $L=\Omega_q(1/\epsilon^2)$ for the list
                  size and upper bound $R=O_q(\epsilon^2)$ for the
                  rate. Previously only existence (and not abundance)
                  of such codes was known for the special case $q=2$
                  (Guruswami, H{\aa}stad, Sudan and Zuckerman, 2002).
                  In order to obtain our result, we employ a relaxed
                  version of the well known Johnson bound on list
                  decoding that translates the \textit{average}
                  Hamming distance between codewords to list decoding
                  guarantees. We furthermore prove that the desired
                  average-distance guarantees hold for a code provided
                  that a natural complex matrix encoding the codewords
                  satisfies the Restricted Isometry Property with
                  respect to the Euclidean norm (RIP-2). For the case
                  of random binary linear codes, this matrix coincides
                  with a random submatrix of the Hadamard-Walsh
                  transform matrix that is well studied in the
                  compressed sensing literature.  Finally, we improve
                  the analysis of Rudelson and Vershynin (2008) on the
                  number of random frequency samples required for
                  exact reconstruction of $k$-sparse signals of length
                  $N$. Specifically, we improve the number of samples
                  from $O(k \log(N) \log^2(k) (\log k + \log\log N))$
                  to $O(k \log(N) \cdot \log^3(k))$. The proof
                  involves bounding the expected supremum of a related
                  Gaussian process by using an improved analysis of
                  the metric defined by the process. This improvement
                  is crucial for our application in list decoding.},
  note =	 {Preliminary version in Proceedings of {SODA 2013.}},
  url_Paper =	 {https://eccc.weizmann.ac.il//report/2012/082}
}

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