Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery

Tony Cai and Anru Zhang

---

• Abstract: This paper establishes a sharp condition on the restrict isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix A satisfies the RIP condition δ^A_k <1/3, then all k-sparse signals β can be recovered exactly via the constrained l₁ minimization based on y = Aβ. Similarly, if the linear map M satisfies the RIP condition δ^M_r <1/3, then all matrices X of rank at most r can be recovered exactly via the constrained nuclear norm minimization based on b = M(X). Furthermore, in both cases it is not possible to do so in general when the condition does not hold. In addition, noisy cases are considered and oracle inequalities are given under the sharp RIP condition.

• Other related papers:

  Cai, T., Wang, L. & Xu, G. (2010).
  New Bounds for Restricted Isometry Constants.
  IEEE Transactions on Information Theory 56 , 4388-4394.

  Cai, T., Wang, L. & Xu, G. (2010).
  Stable recovery of sparse signals and an oracle inequality.
  IEEE Transactions on Information Theory 56 , 3516-3522.

  Cai, T., Wang, L. & Xu, G. (2010).
  Shifting inequality and recovery of sparse signals.
  IEEE Transactions on Signal Processing 58, 1300-1308.

  Cai, T., Xu, G. & Zhang, J. (2009).
  On recovery of sparse signals via l₁ minimization.
  IEEE Transactions on Information Theory 55 , 3388-3397.