Skip to main content
SpringerLink
Log in

The Structure of Minimizers of the Frame Potential on Fusion Frames

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

In this paper we study the fusion frame potential that is a generalization of the Benedetto-Fickus (vectorial) frame potential to the finite-dimensional fusion frame setting. We study the structure of local and global minimizers of this potential, when restricted to suitable sets of fusion frames. These minimizers are related to tight fusion frames as in the classical vector frame case. Still, tight fusion frames are not as frequent as tight frames; indeed we show that there are choices of parameters involved in fusion frames for which no tight fusion frame can exist. We exhibit necessary and sufficient conditions for the existence of tight fusion frames with prescribed parameters, involving the so-called Horn-Klyachko’s compatibility inequalities. The second part of the work is devoted to the study of the minimization of the fusion frame potential on a fixed sequence of subspaces, with a varying sequence of weights. We related this problem to the index of the Hadamard product by positive matrices and use it to give different characterizations of these minima.

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

Access this article

Log in via an institution

Price includes VAT (Finland)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ando, T.: Majorization, doubly stochastic matrices and comparison of eigenvalues. Lecture Notes, Hokkaido Univ. (1982)

  2. Benedetto, J.J., Fickus, M.: Finite normalized tight frames. Adv. Comput. Math. 18(2–4), 357–385 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics, vol. 169. Springer, New York (1996)

    MATH  Google Scholar 

  4. Bodmann, B.G.: Optimal linear transmission by loss-insensitive packet encoding. Appl. Comput. Harmon. Anal. 22(3), 274–285 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bodmann, B.G., Kribs, D.W., Paulsen, V.I.: Decoherence-insensitive quantum communication by optimal C *-encoding. IEEE Trans. Inf. Theory 53(12), 4738–4749 (2007)

    Article  MathSciNet  Google Scholar 

  6. Casazza, P.G., Kutyniok, G.: Frames of subspaces. Contemp. Math. 345, 87–113 (2004)

    MathSciNet  Google Scholar 

  7. Casazza, P.G., Kutyniok, G.: Robustness of fusion frames under erasures of subspaces and of local frame vectors. Contemp. Math. 464, 149–160 (2008)

    MathSciNet  Google Scholar 

  8. Casazza, P.G., Fickus, M.: Minimizing fusion frame potential. Acta Appl. Math. 107(1–3), 7–24 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Casazza, P.G., Fickus, M., Kovacevic, J., Leon, M.T., Tremain, J.C.: A physical interpretation of tight frames. In: Harmonic Analysis and Applications. Appl. Numer. Harmon. Anal., pp. 51–76. Birkhäuser Boston, Boston (2006)

    Chapter  Google Scholar 

  10. Casazza, P.G., Kutyniok, G., Li, S., Rozell, C.J.: Modeling sensor networks with fusion frames. Proc. SPIE 6701, 67011M-1–67011M-11 (2007)

    Google Scholar 

  11. Casazza, P.G., Kutyniok, G., Li, S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25(1), 114–132 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Corach, G., Stojanoff, D.: Index of Hadamard multiplication by positive matrices II. Linear Algebra Appl. 332–334(1–3), 503–517 (2001)

    Article  MathSciNet  Google Scholar 

  13. Fulton, W.: Young Tableaux: With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  14. Fulton, W.: Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Am. Math. Soc. New Ser. 37(3), 209–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Klyachko, A.: Stable bundles, representation theory and Hermitian operators. Sel. Math. New Ser. 4(3), 419–445 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kutyniok, G., Pezeshki, A., Calderbank, R., Liu, T.: Robust dimension reduction, fusion frames, and Grassmannian packings. Appl. Comput. Harmon. Anal. 26(1), 64–76 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Massey, P.: Optimal reconstruction systems for erasures and for the q-potential. Linear Algebra Appl. 431(8), 1302–1316 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Massey, P., Ruiz, M.: Minimization of convex functionals over frame operators. Adv. Comput. Math. doi:10.1007/s10444-008-9092-5

  19. Rozell, C.J., Johnson, D.H.: Analyzing the robustness of redundant population codes in sensory and feature extraction systems. Neurocomputing 69, 1215–1218 (2006)

    Article  Google Scholar 

  20. Ruiz, M., Stojanoff, D.: Some properties of frames of subspaces obtained by operator theory methods. J. Math. Anal. Appl. 343(1), 366–378 (2008)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Depto. de Matemática, FCE-UNLP, La Plata, Argentina

    Pedro G. Massey, Mariano A. Ruiz & Demetrio Stojanoff

  2. IAM-CONICET, Buenos Aires, Argentina

    Pedro G. Massey, Mariano A. Ruiz & Demetrio Stojanoff

Authors
  1. Pedro G. Massey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mariano A. Ruiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Demetrio Stojanoff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pedro G. Massey.

Additional information

Communicated by Peter Casazza.

Partially supported by CONICET (PIP 5272/05) and Universidad de La Plata (UNLP 11 X472).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Massey, P.G., Ruiz, M.A. & Stojanoff, D. The Structure of Minimizers of the Frame Potential on Fusion Frames. J Fourier Anal Appl 16, 514–543 (2010). https://doi.org/10.1007/s00041-009-9098-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-009-9098-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation