Stability analysis of a class of sparse optimization problems

References

• J. Andersson and J.O. Strömberg, On the theorem of uinform recoery of structured random matrices, IEEE Trans. Inform. Theory 60 (2014), pp. 1700–1710. doi: 10.1109/TIT.2014.2300092
   Web of Science ®Google Scholar
• P.T. Boufounos, Greedy sparse signal reconstruction from sign measurements, Proceedings of the 43rd Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 2009, pp. 1305–1309.
   Google Scholar
• J. Cahill, X. Chen, and R. Wang, The gap between the null space property and restricted isometry property, Linear Algebra Appl. 501 (2016), pp. 363–375. doi: 10.1016/j.laa.2016.03.022
   Web of Science ®Google Scholar
• T. Cai and A. Zhang, Sharp RIP bound for sparse signal and low-rank matrix recovery, Appl. Comput. Harmon. Anal. 35 (2013), pp. 74–93. doi: 10.1016/j.acha.2012.07.010
   Web of Science ®Google Scholar
• T. Cai, L. Wang, and G. Xu, New bounds for restricted isometry constants, IEEE Trans. Inform. Theory 56 (2010), pp. 4388–4394. doi: 10.1109/TIT.2010.2054730
   Web of Science ®Google Scholar
• E. Candès, Compressive sampling, Proc. Int. Congr. Math. 3 (2006), pp. 1433–1452.
   Google Scholar
• E. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory 51 (2005), pp. 4203–4215. doi: 10.1109/TIT.2005.858979
   Web of Science ®Google Scholar
• E. Candès and T. Tao, The dantzig selector: Statistical estimation when p is much larger than n, Ann. Statist. 35 (2007), pp. 2313–2351. doi: 10.1214/009053606000001523
   Web of Science ®Google Scholar
• E. Candès, J. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math. 59 (2006), pp. 1207–1223. doi: 10.1002/cpa.20124
   Web of Science ®Google Scholar
• D. Chen and R. Plemmons, Nonnegativity constraints in numerical analysis, in The Birth of Numerical Analysis, A. Bultheel and R. Cools, eds., World Scientific, Singapore, 2009, pp. 109–139.
   Google Scholar
• S. Chen, D. Donoho, and M. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput 20 (1998), pp. 33–61. doi: 10.1137/S1064827596304010
   Web of Science ®Google Scholar
• A. Cohen, W. Dahmen, and R. DeVore, Compressed sensing and best k-term approximation, J. Amer. Math. Soc. 22 (2009), pp. 211–231. doi: 10.1090/S0894-0347-08-00610-3
   Web of Science ®Google Scholar
• D. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (2006), pp. 1289–1306. doi: 10.1109/TIT.2006.871582
   Web of Science ®Google Scholar
• D. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization, Proc. Natl. Acad. Sci. 100 (2003), pp. 2197–2202. doi: 10.1073/pnas.0437847100
   PubMed Web of Science ®Google Scholar
• R. Dudley, Metric entropy of some classes of sets with differentiable boundaries, J. Approx. Theory 10 (1974), pp. 227–236. doi: 10.1016/0021-9045(74)90120-8
   Google Scholar
• Y. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications, Cambridge University Press, New York, NY, 2012.
   Google Scholar
• S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Springer, New York, 2013.
   Google Scholar
• A.J. Hoffman, On approximate solutions of systems of linear inequalities, J. Res. Nat. Bur. Standards 49 (1952), pp. 263–265. doi: 10.6028/jres.049.027
   Google Scholar
• X. Huo and D. Donoho, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inform. Theory 47 (2001), pp. 2845–2862. doi: 10.1109/18.959265
   Web of Science ®Google Scholar
• Q. Mo and S. Li, New bounds on the restricted isometry constant δ2k, Appl. Comput. Harmon. Anal. 31 (2011), pp. 460–468. doi: 10.1016/j.acha.2011.04.005
   Web of Science ®Google Scholar
• S.M. Robinson, Bounds for error in the solution set of a perturbed linear program, Linear Algebra Appl. 6 (1973), pp. 69–81. doi: 10.1016/0024-3795(73)90007-4
   Google Scholar
• G. Tang and A. Nehorai, Performance analysis of sparse recovery based on constrained minimal singular values, IEEE Trans. Signal Process. 59 (2011), pp. 5734–5745. doi: 10.1109/TSP.2011.2164913
   Web of Science ®Google Scholar
• R. Tibshirani, H. Hoefling, and R. Tibshirani, Nearly-isotonic regression, Technometrics 53 (2011), pp. 54–61. doi: 10.1198/TECH.2010.10111
   Web of Science ®Google Scholar
• R. Tibshirani, M. Wainwright, and T. Hastie, Statistical Learning with Sparsity: The Lasso and Generalizations, Chapman and Hall/CRC, Boca Raton, FL, 2015.
   Google Scholar
• A. Tillmann and M. Pfetsch, The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing, IEEE Trans. Inform. Theory 60 (2014), pp. 1248–1259. doi: 10.1109/TIT.2013.2290112
   Google Scholar
• Y. Zhang, A simple proof for recoverability of ℓ1-minimization (ii): The nonnegative case, Tech. Rep., Rice University, 2005.
   Google Scholar
• Y.B. Zhao, RSP-based analysis for sparsest and least ℓ1-norm solutions to underdetermined linear systems, IEEE Trans. Signal Process. 61 (2013), pp. 5777–5788. doi: 10.1109/TSP.2013.2281030
   Google Scholar
• Y.B. Zhao, Equivalence and strong equivalence between the sparsest and least ℓ1-norm nonnegative solutions of linear systems and their applications, J. Oper. Res. Soc. China 2 (2014), pp. 171–193. doi: 10.1007/s40305-014-0043-1
   Google Scholar
• Y.B. Zhao, Sparse Optimization Theory and Methods, CRC Press, Boca Raton, FL, 2018.
   Google Scholar
• Y.B. Zhao and D. Li, Theoretical analysis of sparse recovery stability of Dantzig selector and Lasso, (2017). Available at arXiv: 1711.03783.
   Google Scholar
• Y.B. Zhao and C. Xu, 1-Bit compressive sensing: Reformulation and RRSP-based sign recovery theory, Sci. China Math. 59 (2016), pp. 2049–2074. doi: 10.1007/s11425-016-5153-2
   Web of Science ®Google Scholar
• Y.B. Zhao, H. Jiang, and Z.Q. Luo, Weak stability of ℓ1-minimization methods in sparse data reconstruction, Math. Oper. Res. 44 (2019), pp. 173–195.
   Web of Science ®Google Scholar