Foundations of Computational Mathematics

, Volume 15, Issue 5, pp 1279–1314 | Cite as

Local Convergence of an Algorithm for Subspace Identification from Partial Data

  • Laura Balzano
  • Stephen J. WrightEmail author


Grassmannian rank-one update subspace estimation (GROUSE) is an iterative algorithm for identifying a linear subspace of \(\mathbb {R}^n\) from data consisting of partial observations of random vectors from that subspace. This paper examines local convergence properties of GROUSE, under assumptions on the randomness of the observed vectors, the randomness of the subset of elements observed at each iteration, and incoherence of the subspace with the coordinate directions. Convergence at an expected linear rate is demonstrated under certain assumptions. The case in which the full random vector is revealed at each iteration allows for much simpler analysis and is also described. GROUSE is related to incremental SVD methods and to gradient projection algorithms in optimization.


Subspace identification Optimization Incomplete data 

Mathematics Subject Classification

90C52 65Y20 68W20 



We are grateful to two referees for helpful and constructive comments on the original version of this manuscript.


  1. 1.
    B. A. Ardekani, J. Kershaw, K. Kashikura, and I. Kanno, Activation detection in functional MRIusing subspace modeling and maximum likelihood estimation, IEEETransactions on Medical Imaging, 18 (1999), pp. 101–114.Google Scholar
  2. 2.
    L. Balzano, Handling Missing Data in High-DimensionalSubspace Modeling, PhD thesis, University of Wisconsin-Madison, May2012.Google Scholar
  3. 3.
    L. Balzano, R. Nowak, and B. Recht, Online identification and tracking of subspaces from highlyincomplete information, in 48th Annual Allerton Conference OnCommunication, Control, and Computing (Allerton), September 2010,pp. 704–711. Available at
  4. 4.
    L. Balzano, B. Recht, and R. Nowak, High-dimensional matched subspace detection when data are missing,in Proceedings of the International Symposium on Information Theory,IEEE, June 2010, pp. 1638–1642.Google Scholar
  5. 5.
    L. Balzano and S. J. Wright, On GROUSE andincremental SVD, in Proceedings of the 5th International Workshopon Computational Advances in Multi-Sensor Adaptive Processing(CAMSAP), 2013, pp. 1–4.Google Scholar
  6. 6.
    R. Basri and D. Jacobs, Lambertianreflectance and linear subspaces, IEEE Transactions on PatternAnalysis and Machine Intelligence, 25 (2003), pp. 218–233.Google Scholar
  7. 7.
    E. Candès and J. Romberg, Sparsity andincoherence in compressive sampling, Inverse Problems, 23 (2007),pp. 969–985.Google Scholar
  8. 8.
    J. P. Costeira and T. Kanade,A multibody factorization method for independently movingobjects, International Journal of Computer Vision, 29 (1998), pp.159–179.Google Scholar
  9. 9.
    D. Gross, Recovering low-rank matrices from fewcoefficients in any basis, IEEE Transactions on Information Theory,57 (2011), pp. 1548–1566.Google Scholar
  10. 10.
    J. Gupchup, R. Burns, A. Terzis, and A. Szalay, Model-based event detection in wireless sensornetworks, in Proceedings of the Workshop on Data Sharing andInteroperability (DSI), 2007.Google Scholar
  11. 11.
    Nathan Halko, Per-Gunnar Martinsson, and Joel A Tropp, Finding structure with randomness:Probabilistic algorithms for constructing approximate matrixdecompositions, SIAM Review, 53 (2011), pp. 217–288.Google Scholar
  12. 12.
    H. Krim and M. Viberg, Two decades of arraysignal processing research: the parametric approach, IEEE SignalProcessing Magazine, 13 (1996), pp. 67–94.Google Scholar
  13. 13.
    A. Lakhina, M. Crovella, and C. Diot, Diagnosing network-wide traffic anomalies, in Proceedings ofSIGCOMM, 2004, pp. 219–230.Google Scholar
  14. 14.
    D. Manolakis and G. Shaw, Detectionalgorithms for hyperspectral imaging applications, IEEE SignalProcessing Magazine, 19 (2002), pp. 29–43.Google Scholar
  15. 15.
    J. Nocedal and S. J. Wright, NumericalOptimization, Springer, New York, second ed., 2006.Google Scholar
  16. 16.
    S. Papadimitriou, J. Sun, and C. Faloutsos,Streaming pattern discovery in multiple time-series, inProceedings of the 31st International Conference on Very Large DataBases (VLDB ’05), 2005, pp. 697–708.Google Scholar
  17. 17.
    B. Recht, A simpler approach to matrix completion,Journal of Machine Learning Research, 12 (2011), pp. 3413–3430.Google Scholar
  18. 18.
    G. W. Stewart and J. Sun, Matrix PerturbationTheory, Computer Science and Scientific Computing, Academic Press,New York, 1990.Google Scholar
  19. 19.
    L. Tong and S. Perreau, Multichannel blindidentification: From subspace to maximum likelihood methods,Proceedings of the IEEE, 86 (1998), pp. 1951–1968.Google Scholar
  20. 20.
    P. van Overschee and B. de Moor, SubspaceIdentification for Linear Systems, Kluwer Academic Publishers,Norwell, Massachusetts, 1996.Google Scholar
  21. 21.
    L. Vandenberghe, Convex optimization techniques in systemidentification, in Proceedings of the IFAC Symposium on SystemIdentification, July 2012, pp. 71–76.Google Scholar
  22. 22.
    G. S. Wagner and T. J. Owens, Signaldetection using multi-channel seismic data, Bulletin of theSeismological Society of America, 86 (1996), pp. 221–231.Google Scholar

Copyright information

© SFoCM 2014

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer ScienceUniversity of MichiganAnn ArborUSA
  2. 2.Department of Computer SciencesUniversity of Wisconsin-MadisonMadisonUSA

Personalised recommendations