Skip to main content
SpringerLink
Log in

Symplectic geometry and connectivity of spaces of frames

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have natural symplectic descriptions. Symplectic tools seem well-adapted to addressing a number of important questions about frames; in this paper, we focus on the frame homotopy conjecture posed in 2002 and recently proved by Cahill, Mixon, and Strawn, which says that the space of finite unit norm tight frames is connected. We give a simple symplectic proof of a generalization of the frame homotopy conjecture, showing that spaces of complex frames with arbitrary prescribed norms and frame operators are connected. To spark further investigation, we also suggest a number of fundamental questions in frame theory which seem amenable to a symplectic approach.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexeev, B., Cahill, J., Mixon, D.G.: Full spark frames. J. Fourier Anal. Appl. 18(6), 1167–1194 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14(1), 1–15 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Audin, M.: Torus Actions on Symplectic Manifolds, volume 93 of Progress in Mathematics. Springer, Basel (2012). second revised edition

    Google Scholar 

  4. Balan, R., Casazza, P.G., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20(3), 345–356 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bandeira, A.S., Cahill, J., Mixon, D.G., Nelson, A.A.: Saving phase: injectivity and stability for phase retrieval. Appl. Comput. Harmon. Anal. 37(1), 106–125 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bodmann, B.G., Casazza, P.G.: The road to equal-norm Parseval frames. J. Funct. Anal. 258(2), 397–420 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bolcskei, H., Hlawatsch, F.: Noise reduction in oversampled filter banks using predictive quantization. IEEE Trans. Inf. Theory 47(1), 155–172 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cahill, J., Mixon, D.G., Strawn, N.: Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. SIAM J. Appl. Algebra Geom. 1(1), 38–72 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cahill, J., Strawn, N.: Algebraic geometry and finite frames. In: Casazza, P.G., Kutyniok, G. (eds.) Finite Frames: Theory and Applications, Applied and Numerical Harmonic Analysis, pp 141–170. Birkhäuser, Boston (2013)

  10. Candès, E.J., Eldar, Y.C., Strohmer, T., Voroninski, V.: Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6(1), 199–225 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cannas da Silva, A.: Lectures on Symplectic Geometry, volume 1764 of Lecture Notes in Mathematics. Springer, Berlin (2001)

    MATH  Google Scholar 

  12. Cannas da Silva, A.: Symplectic geometry. In: Dillen, F.J.E., Verstraelen, L.C.A. (eds.) Handbook of Differential Geometry, vol. 2, pp 79–188. Elsevier (2006)

  13. Cantarella, J., Duplantier, B., Shonkwiler, C., Uehara, E.: A fast direct sampling algorithm for equilateral closed polygons. J. Phys. A 49, 275202 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cantarella, J., Shonkwiler, C.: The symplectic geometry of closed equilateral random walks in 3-space. Ann. Appl. Probab. 26(1), 549–596 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Casazza, P.G.: The Kadison–Singer and Paulsen problems in finite frame theory. In: Casazza, P.G., Kutyniok, G. (eds.) Finite Frames: Theory and Applications, Applied and Numerical Harmonic Analysis, pp 381–413. Birkhäuser, Boston (2013)

  16. Casazza, P.G., Fickus, M., Kovačević, J., Leon, M.T., Tremain, J.C.: A physical interpretation of tight frames. In: Heil, C. (ed.) Harmonic Analysis and Applications: In Honor of John J. Benedetto, Applied and Numerical Harmonic Analysis, pp 51–76. Birkhäuser, Boston (2006)

  17. Casazza, P.G., Heinecke, A., Krahmer, F., Kutyniok, G.: Optimally sparse frames. IEEE Trans. Inf. Theory 57(11), 7279–7287 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Casazza, P.G., Kovačević, J.: Equal-norm tight frames with erasures. Adv. Comput. Math. 18(2-4), 387–430 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Casazza, P.G., Kutyniok, G.: Editors. Finite Frames: Theory and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2013)

    Book  MATH  Google Scholar 

  20. Casazza, P.G., Leon, M.T.: Existence and construction of finite frames with a given frame operator. Int. J. Pure Appl. Math. 63(2), 149–157 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Conca, A., Edidin, D., Hering, M., Vinzant, C.: An algebraic characterization of injectivity in phase retrieval. Appl. Comput. Harmon. Anal. 38(2), 346–356 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27(5), 1271–1283 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72(2), 341–366 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dykema, K., Strawn, N.: Manifold structure of spaces of spherical tight frames. Int. J. Pure Appl. Math. 28(2), 217–256 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Ehler, M., Hering, M., Manon, C., Needham, T., Shonkwiler, C.: The Paulsen problem made symplectic. Oberwolfach Rep. 15(4), 2790–2794 (2018)

    Google Scholar 

  26. Ehresmann. C.: Sur la topologie de certains espaces homogènes. Ann. Math., Second Series 35(2), 396–443 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  27. Flaschka, H., Millson, J.J.: Bending flows for sums of rank one matrices. Can. J. Math. 57(1), 114–158 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fulton, W.: Young Tableaux, with Applications to Representation Theory and Geometry, volume 35 of London Mathematical Society Student Texts. Cambridge University Press (1997)

  29. Giol, J., Kovalev, L.V., Larson, D., Nguyen, N., Tener, J.E.: Projections and idempotents with fixed diagonal and the homotopy problem for unit tight frames. Oper. Matrices 5(1), 139–155 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Goyal, V.K., Kovačević, J., Kelner, J.A.: Quantized frame expansions with erasures. Appl. Comput. Harmon. Anal. 10(3), 203–233 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hamilton, L., Moitra, A.: The Paulsen problem made simple. In: 10th Innovations in Theoretical Computer Science (ITCS 2019), volume 124 of Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl–Leibniz-Zentrum für Informatik. Dagstuhl, Germany (2018)

  32. Harada, M., Holm, T.S., Jeffrey, L.C., Mare, A.-L.: Connectivity properties of moment maps on based loop groups. Geom. Topol. 10(3), 1607–1634 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Holmes, R.B., Paulsen, V.I.: Optimal frames for erasures. Linear Algebra Appl. 377, 31–51 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Horn, A.: Doubly stochastic matrices and the diagonal of a rotation matrix. Am. J. Math. 76(3), 620–630 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kapovich, M., Millson, J.J.: On the moduli space of polygons in the Euclidean plane. J. Differ. Geom. 42(1), 133–164 (1995)

    MathSciNet  MATH  Google Scholar 

  36. Knutson, A.: The symplectic and algebraic geometry of Horn’s problem. Linear Algebra Appl. 319(1–3), 61–81 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kovačević, J., Chebira, A.: Life beyond bases: the advent of frames (Part I). IEEE Signal Process. Mag. 24(4), 86–104 (2007)

    Article  Google Scholar 

  38. Kovačević, J., Chebira, A.: Life beyond bases: the advent of frames (Part II). IEEE Signal Process. Mag. 24(5), 115–125 (2007)

    Article  Google Scholar 

  39. Kwok, T.C., Lau, L.C., Lee, Y.T., Ramachandran, A.: The Paulsen problem, continuous operator scaling, and smoothed analysis. In: Diakonikolas, I., Kempe, D., Henzinger, M. (eds.) STOC’18—Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pp. 182–189. Association for Computing Machinery, New York (2018)

  40. Lerman, E.: Gradient flow of the norm squared of a moment map. L’Enseignement Mathématique 51, 117–127 (2005)

    MathSciNet  MATH  Google Scholar 

  41. Łojasiewicz, S.: Triangulation of semi-analytic sets. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 18(4), 449–474 (1964)

    MathSciNet  MATH  Google Scholar 

  42. Malikiosis, R.-D.: Spark deficient Gabor frames. Pac. J. Math. 294(1), 159–180 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Manetti, M.: Topology, volume 91 of Unitext, translated by Simon G. Chiossi. Springer, Cham (2015)

    Google Scholar 

  44. Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1), 121–130 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  45. McDuff, D., Salamon, D.: Introduction to Symplectic Topology, 3rd edn. Oxford University Press, Oxford (2017)

    Book  MATH  Google Scholar 

  46. Meyer, K.R.: Symmetries and integrals in mechanics. In: Peixoto, M.M. (ed.) Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26–August 14, 1971, pp. 259–272. Academic Press, New York (1973)

  47. Monk, D.: The geometry of flag manifolds. Proc. Lond. Math. Soc. 9(2), 253–286 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rupf, M., Massey, J.L.: Optimum sequence multisets for synchronous code-division multiple-access channels. IEEE Trans. Inf. Theory 40(4), 1261–1266 (1994)

    Article  MATH  Google Scholar 

  49. Schur, I.: Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungsberichte der Berliner Mathematischen Gesellschaft 22, 9–20 (1923)

    MATH  Google Scholar 

  50. Shonkwiler, C.: The geometry of constrained random walks and an application to frame theory. In: IEEE Statistical Signal Processing Workshop (SSP), pp. 343–347. IEEE (2018)

  51. Sjamaar, R., Lerman, E.: Stratified symplectic spaces and reduction. Ann. Math., Second Series 134(2), 375–422 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  52. Smith, K.: Connectivity and convexity properties of the momentum map for group actions on hilbert manifolds. PhD thesis, University of Toronto. arXiv:1407.4351[math.SG] (2013)

  53. Vietoris, L.: Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen. Math. Ann. 97, 454–472 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  54. Vinzant, C.: A small frame and a certificate of its injectivity. In: 2015 International Conference on Sampling Theory and Applications (SampTA), pp. 197–200. IEEE (2015)

  55. Viswanath, P., Anantharam, V.: Optimal sequences and sum capacity of synchronous CDMA systems. IEEE Trans. Inf. Theory 45(6), 1984–1991 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  56. Viswanath, P., Anantharam, V.: Optimal sequences for CDMA under colored noise: a Schur-saddle function property. IEEE Trans. Inf. Theory 48(6), 1295–1318 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  57. Wansbrough, D.: What are flag manifolds and why are they interesting? Austral. Math. Soc. Gaz. 24(4), 150–153 (1997)

    MathSciNet  MATH  Google Scholar 

  58. Ye, K., Wong, K.S.-W., Lim, L.-H.: Optimization on flag manifolds. Preprint, arXiv:1907.00949 [math.OC] (2019)

  59. Zimmermann, G.: Normalized tight frames in finite dimensions. In: Haussmann, W., Jetter, K., Reimer, M. (eds.) Recent Progress in Multivariate Approximation: 4th International Conference, Witten-Bommerholz (Germany), September 2000, volume 137 of ISNM International Series of Numerical Mathematics, pp. 249–252. Birkhäuser, Basel (2001)

Download references

Acknowledgments

This project grew out of a conversation we had at the CMO–BIRS Workshop on the Geometry and Topology of Knotting and Entanglement in Proteins in November, 2017, and we would like to thank the organizers, BIRS, and the Casa Matemática Oaxaca for a very stimulating workshop. Conversations we had with virtually all the participants at the Oberwolfach Mini-Workshop on Algebraic, Geometric, and Combinatorial Methods in Frame Theory in October, 2018 significantly refined our thinking and opened our eyes to broader applications of our symplectic ideas, so we would also like to thank the organizers and all the participants as well as the Mathematisches Forschungsinstitut Oberwolfach. We are very grateful for ongoing conversations about frames with various friends and colleagues, especially Jason Cantarella, Martin Ehler, Simon Foucart, Milena Hering, Emily King, Chris Manon, Dustin Mixon, Louis Scharf, and Nate Strawn, and we thank the anonymous referees for their helpful suggestions and ideas which have improved both the exposition in this paper and our own understanding of these ideas.

Funding

This work was supported by a grant from the Simons Foundation (#354225, Clayton Shonkwiler).

Author information

Authors and Affiliations

  1. Department of Mathematics, Florida State University, Tallahassee, FL, USA

    Tom Needham

  2. Department of Mathematics, Colorado State University, Fort Collins, CO, USA

    Clayton Shonkwiler

Authors
  1. Tom Needham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Clayton Shonkwiler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tom Needham.

Additional information

Communicated by: Felix Krahmer

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Needham, T., Shonkwiler, C. Symplectic geometry and connectivity of spaces of frames. Adv Comput Math 47, 5 (2021). https://doi.org/10.1007/s10444-020-09842-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-020-09842-7

Keywords

Mathematics subject classification (2010)

Search

Navigation