SpringerLink
Log in

Recovery Under Side Constraints

  • Chapter
  • First Online:
Compressed Sensing in Information Processing

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

  • 634 Accesses

Abstract

This chapter addresses sparse signal reconstruction under various types of structural side constraints with applications in multi-antenna systems. Side constraints may result from prior information on the measurement system and the sparse signal structure. They may involve the structure of the sensing matrix, the structure of the non-zero support values, the temporal structure of the sparse representation vector, and the nonlinear measurement structure. First, we demonstrate how a priori information in the form of structural side constraints influence recovery guarantees (null space properties) using ℓ 1-minimization. Furthermore, for constant modulus signals, signals with row, block, and rank sparsity, as well as non-circular signals, we illustrate how structural prior information can be used to devise efficient algorithms with improved recovery performance and reduced computational complexity. Finally, we address the measurement system design for linear and nonlinear measurements of sparse signals. To this end, we derive a new linear mixing matrix design based on coherence minimization. Then, we extend our focus to nonlinear measurement systems where we design parallel optimization algorithms to efficiently compute stationary points in the sparse phase-retrieval problem with and without dictionary learning.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 117.69
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 149.79
Price includes VAT (Germany)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
EUR 149.79
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Let U ΛU H be the eigenvalue decomposition of A H A. Then, the unconstrained mixing matrix is obtained as \(\boldsymbol {\Phi }_{\text{uncon}} = {\boldsymbol \Lambda }^{-1/2}_N {\mathbf {U}}^{\mathrm {H}}_N\), where Λ N and U N contain the leading N eigenvalues and eigenvectors, respectively. For constrained mixing matrix scenarios, simply Φ con =  Π( Φ uncon).

References

  1. Abolghasemi, V., Ferdowsi, S., Makkiabadi, B., Sanei, S.: On optimization of the measurement matrix for compressive sensing. In: Proceedings of the 18th European Signal Processing Conference, pp. 427–431 (2010)

    Google Scholar 

  2. Ardah, K., d. Almeida, A.L.F., Haardt, M.: Low-complexity millimeter wave CSI estimation in MIMO-OFDM hybrid beamforming systems. In: WSA 2019; 23rd International ITG Workshop on Smart Antennas, pp. 1–5 (2019)

    Google Scholar 

  3. Ardah, K., de Almeida, A.L.F., Haardt, M.: A gridless CS approach for channel estimation in hybrid massive MIMO systems. In: 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4160–4164 (2019)

    Google Scholar 

  4. Ardah, K., Pesavento, M., Haardt, M.: A novel sensing matrix design for compressed sensing via mutual coherence minimization. In: 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), pp. 66–70 (2019)

    Google Scholar 

  5. Ardah, K., Sokal, B., de Almeida, A.L.F., Haardt, M.: Compressed sensing based channel estimation and open-loop training design for hybrid analog-digital massive MIMO systems. In: 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4597–4601 (2020)

    Google Scholar 

  6. Ardah, K., Gherekhloo, S., de Almeida, A.L.F., Haardt, M.: TRICE: A channel estimation framework for RIS-aided millimeter-wave MIMO systems. IEEE Signal Process. Lett. 28, 513–517 (2021)

    Article  Google Scholar 

  7. Boyer, R., Haardt, M.: Noisy compressive sampling based on block-sparse tensors: Performance limits and beamforming techniques. IEEE Trans. Signal Process. (23), 6075–6088 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choi, J.W., Shim, B., Ding, Y., Rao, B., Kim, D.I.: Compressed sensing for wireless communications: Useful tips and tricks. IEEE Commun. Surv. Tutorials 19(3), 1527–1550 (2017)

    Article  Google Scholar 

  9. Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fannjiang, A., Strohmer, T.: The numerics of phase retrieval. Acta Numerica 29, 125–228 (2020)

    Article  MathSciNet  Google Scholar 

  11. Fischer, T., Pfetsch, M.E.: Monoidal cut strengthening and generalized mixed-integer rounding for disjunctive programs. Oper. Res. Lett. 45(6), 556–560 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fischer, T., Pfetsch, M.E.: Branch-and-cut for linear programs with overlap** SOS1 constraints. Math. Prog. Comp. 10(1), 33–68 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fischer, T., Hegde, G., Matter, F., Pesavento, M., Pfetsch, M.E., Tillmann, A.M.: Joint antenna selection and phase-only beamforming using mixed-integer nonlinear programming. In: WSA 2018; 22nd International ITG Workshop on Smart Antennas, pp. 1–7 (2018)

    Google Scholar 

  14. Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York (2013)

    Google Scholar 

  15. Gao, F., Tian, Z., Larsson, E.G., Pesavento, M., **, S.: Introduction to the special issue on array signal processing for angular models in massive MIMO communications. IEEE J. Sel. Topics Signal Process. 13(5), 882–885 (2019)

    Article  Google Scholar 

  16. Gribonval, R., Nielsen, M.: Sparse representations in unions of bases. IEEE Trans. Inf. Theory 49(12), 3320–3325 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Haardt, M., Roemer, F.: Enhancements of Unitary ESPRIT for non-circular sources. In: 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Montreal, vol. II, pp. 101–104 (2004)

    Google Scholar 

  18. Haardt, M., Pesavento, M., Roemer, F., El Korso, M.N.: Subspace methods and exploitation of special array structures. In: Zoubir, A.M., Viberg, M., Chellappa, R., Theodoridis, S. (eds.) Academic Press Library in Signal Processing: Volume 3 – Array and Statistical Signal Processing, pp. 651–717. Elsevier, Amsterdam (2014). Chapter 15

    Chapter  Google Scholar 

  19. Hand, P., Voroninski, V.: Compressed sensing from phaseless Gaussian measurements via linear programming in the natural parameter space. Preprint, ar**v:1611.05985 (2016)

    Google Scholar 

  20. Hegde, G., Yang, Y., Steffens, C., Pesavento, M.: Parallel low-complexity M-PSK detector for large-scale MIMO systems. In: 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM), pp. 1–5. IEEE, Piscataway (2016)

    Google Scholar 

  21. Hegde, G., Pesavento, M., Pfetsch, M.E.: Joint active device identification and symbol detection using sparse constraints in massive MIMO systems. In: 2017 25th European Signal Processing Conference (EUSIPCO), pp. 703–707. IEEE, Piscataway (2017)

    Google Scholar 

  22. Heuer, J., Matter, F., Pfetsch, M.E., Theobald, T.: Block-sparse recovery of semidefinite systems and generalized null space conditions. Linear Algebra Appl. 603, 470–495 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hyder, M.M., Mahata, K.: Direction-of-arrival estimation using a mixed ℓ 2,0 norm approximation. IEEE Trans. Signal Process. 58(9), 4646–4655 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jaganathan, K., Oymak, S., Hassibi, B.: Sparse phase retrieval: uniqueness guarantees and recovery algorithms. IEEE Trans. Signal Process. 65(9), 2402–2410 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Juditsky, A., Karzan, F.K., Nemirovski, A.: On a unified view of nullspace-type conditions for recoveries associated with general sparsity structures. Linear Algebra Appl. 441, 124–151 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Keiper, S., Kutyniok, G., Lee, D.G., Pfander, G.E.: Compressed sensing for finite-valued signals. Linear Algebra Appl. 532, 570–613 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Khajehnejad, M.A., Dimakis, A.G., Xu, W., Hassibi, B.: Sparse recovery of nonnegative signals with minimal expansion. IEEE Trans. Signal Process. 59(1), 196–208 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kong, L., Sun, J., **u, N.: S-semigoodness for low-rank semidefinite matrix recovery. Pac. J. Optim. 10(1), 73–83 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Kowalski, M.: Sparse regression using mixed norms. Appl. Comput. Harmon. Anal. 27(3), 303–324 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Krim, H., Viberg, M.: Two decades of array signal processing research: the parametric approach. IEEE Signal Process. Mag. 13(4), 67–94 (1996)

    Article  Google Scholar 

  31. Kuske, J., Swoboda, P., Petra, S.: A novel convex relaxation for non-binary discrete tomography. In: International Conference on Scale Space and Variational Methods in Computer Vision, pp. 235–246. Springer, Berlin (2017)

    Google Scholar 

  32. Kushe, G., Yang, Y., Steffens, C., Pesavento, M.: A parallel sparse regularization method for structured multilinear low-rank tensor decomposition. In: 2019 27th European Signal Processing Conference (EUSIPCO), pp. 1–5 (2019)

    Google Scholar 

  33. Kushe, G., Yang, Y., Pesavento, M.: A block successive convex approximation framework for multidimensional harmonic retrieval and imperfect measurements. In: WSA 2020; 24th International ITG Workshop on Smart Antennas, pp. 1–5 (2020)

    Google Scholar 

  34. Lange, J.H., Pfetsch, M.E., Seib, B.M., Tillmann, A.M.: Sparse recovery with integrality constraints. Discrete Applied Math. 283, 346–366 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, X., Voroninski, V.: Sparse signal recovery from quadratic measurements via convex programming. SIAM J. Math. Anal. 45(5), 3019–3033 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, X., Ye, J., Li, G., Bai, H., Jiang, Q.: A new approach to sensing matrix optimization using steepest descent algorithm. In: 2015 34th Chinese Control Conference (CCC), pp. 4939–4944 (2015)

    Google Scholar 

  37. Liu, T., Hoang, M.T., Yang, Y., Pesavento, M.: A block coordinate descent algorithm for sparse Gaussian graphical model inference with Laplacian constraints. In: 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), pp. 236–240 (2019)

    Google Scholar 

  38. Liu, T., Hoang, M.T., Yang, Y., Pesavento, M.: A parallel optimization approach on the infinity norm minimization problem. In: 2019 27th European Signal Processing Conference (EUSIPCO), pp. 1–5. IEEE, Piscataway (2019)

    Google Scholar 

  39. Liu, T., Tillmann, A.M., Yang, Y., Eldar, Y.C., Pesavento, M.: A parallel algorithm for phase retrieval with dictionary learning. In: 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2021)

    Google Scholar 

  40. Lu, C., Li, H., Lin, Z.: Optimized projections for compressed sensing via direct mutual coherence minimization. Signal Process. 151, 45–55 (2018)

    Article  Google Scholar 

  41. Malioutov, D., Çetin, M., Willsky, A.: A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Trans. Signal Process. 53(8), 3010–3022 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Netrapalli, P., Jain, P., Sanghavi, S.: Phase retrieval using alternating minimization. IEEE Trans. Signal Process. 63(18), 4814–4826 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ohlsson, H., Eldar, Y.C.: On conditions for uniqueness in sparse phase retrieval. In: 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1841–1845 (2014)

    Google Scholar 

  44. Ohlsson, H., Yang, A., Dong, R., Sastry, S.: CPRL – an extension of compressive sensing to the phase retrieval problem. In: Pereira, F., Burges, C.J.C., Bottou, L., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, pp. 1367–1375. Curran Associates Inc., Red Hook (2012)

    Google Scholar 

  45. Oymak, S., Hassibi, B.: New null space results and recovery thresholds for matrix rank minimization. In: Proceedings of the ISIT 2011. Preprint. ar**v:1011.6326 (2010)

    Google Scholar 

  46. Park, J., Lee, G., Sung, Y., Yukawa, M.: Coordinated beamforming with relaxed zero forcing: the sequential orthogonal projection combining method and rate control. IEEE Trans. Signal Process. 61(12), 3100–3112 (2013)

    Article  Google Scholar 

  47. Qiu, T., Palomar, D.P.: Undersampled sparse phase retrieval via majorization-minimization. IEEE Trans. Signal Process. 65(22), 5957–5969 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rani, M., Dhok, S.B., Deshmukh, R.B.: A systematic review of compressive sensing: concepts, implementations and applications. IEEE Access 6, 4875–4894 (2018)

    Article  Google Scholar 

  49. Shechtman, Y., Beck, A., Eldar, Y.C.: Gespar: Efficient phase retrieval of sparse signals. IEEE Trans. Signal Process. 62(4), 928–938 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. Steffens, C., Pesavento, M.: Block- and rank-sparse recovery for direction finding in partly calibrated arrays. IEEE Trans. Signal Process. 66(2), 384–399 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  51. Steffens, C., Pesavento, M.: Collaborative Sensing Techniques, chap. 7, pp. 121–145. John Wiley & Sons Ltd., Hoboken (2020)

    Google Scholar 

  52. Steffens, C., Yang, Y., Pesavento, M.: Multidimensional sparse recovery for MIMO channel parameter estimation. In: 2016 24th European Signal Processing Conference (EUSIPCO), pp. 66–70 (2016)

    Google Scholar 

  53. Steffens, C., Suleiman, W., Sorg, A., Pesavento, M.: Gridless compressed sensing under shift-invariant sampling. In: 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4735–4739 (2017)

    Google Scholar 

  54. Steffens, C., Pesavento, M., Pfetsch, M.E.: A compact formulation for the ℓ 2,1 mixed-norm minimization problem. IEEE Trans. Signal Process. 66(6), 1483–1497 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  55. Steinwandt, J., Roemer, F., Haardt, M.: Sparsity-based direction-of-arrival estimation for strictly non-circular sources. In: 2016 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Shanghai (2016)

    Google Scholar 

  56. Steinwandt, J., Roemer, F., Haardt, M., Del Galdo, G.: Deterministic Cramér-Rao bound for strictly non-circular sources and analytical analysis of the achievable gains. IEEE Trans. Signal Process. 64(17), 4417–4431 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  57. Steinwandt, J., Roemer, F., Steffens, C., Haardt, M., Pesavento, M.: Gridless superresolution direction finding for strictly non-circular sources based on atomic norm minimization. In: 2016 50th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove (2016)

    Google Scholar 

  58. Steinwandt, J., Steffens, C., Pesavento, M., Haardt, M.: Sparsity-aware direction finding for strictly non-circular sources based on rank minimization. In: 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM), Rio de Janeiro (2016)

    Google Scholar 

  59. Steinwandt, J., Roemer, F., Haardt, M.: Generalized least squares for ESPRIT-type direction of arrival estimation. IEEE Signal Process. Lett. 24(11), 1681–1685 (2017)

    Article  Google Scholar 

  60. Steinwandt, J., Roemer, F., Haardt, M.: Performance analysis of ESPRIT-type algorithms for co-array structures. In: 2017 IEEE 7th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), pp. 1–5 (2017)

    Google Scholar 

  61. Steinwandt, J., Roemer, F., Haardt, M., Del Galdo, G.: Performance analysis of multi-dimensional ESPRIT-type algorithms for arbitrary and strictly non-circular sources with spatial smoothing. IEEE Trans. Signal Process. 65(9), 2262–2276 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  62. Stojnic, M.: Recovery thresholds for ℓ 1 optimization in binary compressed sensing. In: 2010 IEEE International Symposium on Information Theory, pp. 1593–1597. IEEE, Piscataway (2010)

    Google Scholar 

  63. Stojnic, M., Parvaresh, F., Hassibi, B.: On the reconstruction of block-sparse signals with an optimal number of measurements. IEEE Trans. Signal Process. 57(8), 3075–3085 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  64. Suleiman, W., Steffens, C., Sorg, A., Pesavento, M.: Gridless compressed sensing for fully augmentable arrays. In: 2017 25th European Signal Processing Conference (EUSIPCO), pp. 1986–1990 (2017)

    Google Scholar 

  65. Tillmann, A.M., Pfetsch, M.E.: The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing. IEEE Trans. Inf. Theory 60(2), 1248–1259 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  66. Tillmann, A.M., Eldar, Y.C., Mairal, J.: DOLPHIn – dictionary learning for phase retrieval. IEEE Trans. Signal Process. 64(24), 6485–6500 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  67. Tropp, J.A.: Algorithms for simultaneous sparse approximation. Part II: convex relaxation. Signal Process. 86(3), 589–602 (2006)

    MATH  Google Scholar 

  68. Tropp, J.A., Dhillon, I.S., Heath, R.W., Strohmer, T.: Designing structured tight frames via an alternating projection method. IEEE Trans. Inf. Theory 51(1), 188–209 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  69. Turlach, B.A., Venables, W.N., Wright, S.J.: Simultaneous variable selection. Technometrics 47(3), 349–363 (2005)

    Article  MathSciNet  Google Scholar 

  70. Van Trees, H.L.: Optimum Array Processing. Wiley, New York (2002)

    Book  Google Scholar 

  71. Vigerske, S.: Decomposition in multistage stochastic programming and a constraint integer programming approach to mixed-integer nonlinear programming. Ph.D. Thesis, Humboldt-Universität zu Berlin (2013)

    Google Scholar 

  72. Walewski, A.C., Steffens, C., Pesavento, M.: Off-grid parameter estimation based on joint sparse regularization. In: SCC 2017; 11th International ITG Conference on Systems, Communications and Coding, pp. 1–6 (2017)

    Google Scholar 

  73. Wang, G., Zhang, L., Giannakis, G.B., Akçakaya, M., Chen, J.: Sparse phase retrieval via truncated amplitude flow. IEEE Trans. Signal Process. 66(2), 479–491 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  74. Wang, X., Liu, T., Trinh-Hoang, M., Pesavento, M.: GPU-accelerated parallel optimization for sparse regularization. In: 2020 IEEE 11th Sensor Array and Multichannel Signal Processing Workshop (SAM), pp. 1–5 (2020)

    Google Scholar 

  75. Yang, Y., Pesavento, M.: A unified successive pseudoconvex approximation framework. IEEE Trans. Signal Process. 65(13), 3313–3328 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  76. Yang, Y., Pesavento, M.: Energy efficiency in MIMO interference channels: social optimality and max-min fairness. In: 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3689–3693 (2018)

    Google Scholar 

  77. Yang, Y., Pesavento, M.: A parallel best-response algorithm with exact line search for nonconvex sparsity-regularized rank minimization. In: 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6323–6327 (2018)

    Google Scholar 

  78. Yang, Y., Pesavento, M., Zhang, M., Palomar, D.P.: An online parallel algorithm for recursive estimation of sparse signals. IEEE Trans. Signal Inf. Process. Netw. 2(3), 290–305 (2016)

    MathSciNet  Google Scholar 

  79. Yang, Y., Pesavento, M., Chatzinotas, S., Ottersten, B.: Parallel and hybrid soft-thresholding algorithms with line search for sparse nonlinear regression. In: European Signal Processing Conference, vol. 2018, pp. 1587–1591 (2018)

    Google Scholar 

  80. Yang, Y., Pesavento, M., Chatzinotas, S., Ottersten, B.: Successive convex approximation algorithms for sparse signal estimation with nonconvex regularizations. IEEE J. Sel. Topics Signal Process. 12(6), 1286–1302 (2018)

    Article  Google Scholar 

  81. Yang, Y., Pesavento, M., Chatzinotas, S., Ottersten, B.: Energy efficiency optimization in MIMO interference channels: a successive pseudoconvex approximation approach. IEEE Trans. Signal Process. 67(15), 4107–4121 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  82. Yang, Y., Pesavento, M., Eldar, Y.C., Ottersten, B.: Parallel coordinate descent algorithms for sparse phase retrieval. In: 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 7670–7674 (2019)

    Google Scholar 

  83. Yang, Y., Pesavento, M., Luo, Z.Q., Ottersten, B.: Inexact block coordinate descent algorithms for nonsmooth nonconvex optimization. IEEE Trans. Signal Process. 68, 947–961 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  84. Yu, L., Li, G., Chang, L.: Optimizing projection matrix for compressed sensing systems. In: 2011 8th International Conference on Information, Communications Signal Processing (ICICS), pp. 1–5 (2011)

    Google Scholar 

  85. Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. Series B (Statistical Methodology) 68(1), 49–67 (2006)

    Google Scholar 

  86. Zelnik-Manor, L., Rosenblum, K., Eldar, Y.C.: Sensing matrix optimization for block-sparse decoding. IEEE Trans. Signal Process. 59(9), 4300–4312 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  87. Zhang, Y.: A simple proof for recoverability of ℓ 1-minimization (II): the nonnegativity case. Technical report TR05-10, Dept. of Computational and Applied Mathematics, Rice University (2005)

    Google Scholar 

  88. Liu, T., Tillmann, A.M., Yang, Y., Eldar, Y.C., Pesavento, M.: Extended Successive Convex Approximation for Phase Retrieval with Dictionary Learning. Preprint, ar**v:2109.05646 (2022)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Communications Research Laboratory, Ilmenau University of Technology, Ilmenau, Germany

    Khaled Ardah & Martin Haardt

  2. Institut für Nachrichtentechnik, TU Darmstadt, Darmstadt, Germany

    Tianyi Liu & Marius Pesavento

  3. Department of Mathematics, TU Darmstadt, Darmstadt, Germany

    Frederic Matter & Marc E. Pfetsch

Authors
  1. Khaled Ardah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Haardt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tianyi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Frederic Matter
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Marius Pesavento
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Marc E. Pfetsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marius Pesavento .

Editor information

Editors and Affiliations

  1. Mathematisches Institut, Ludwig Maximilian University of Munich, München, Bayern, Germany

    Gitta Kutyniok

  2. Lehrstuhl für Mathematik, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Holger Rauhut

  3. Lehrstuhl für Mathematik, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Robert J. Kunsch

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ardah, K., Haardt, M., Liu, T., Matter, F., Pesavento, M., Pfetsch, M.E. (2022). Recovery Under Side Constraints. In: Kutyniok, G., Rauhut, H., Kunsch, R.J. (eds) Compressed Sensing in Information Processing. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-09745-4_7

Download citation

Publish with us

Policies and ethics

Search

Navigation