Letters in Mathematical Physics

, Volume 109, Issue 8, pp 1777–1804 | Cite as

Matrix versions of the Hellinger distance

  • Rajendra Bhatia
  • Stephane Gaubert
  • Tanvi JainEmail author


On the space of positive definite matrices, we consider distance functions of the form \(d(A,B)=\left[ \mathrm{tr}\mathcal {A}(A,B)-\mathrm{tr}\mathcal {G}(A,B)\right] ^{1/2},\) where \(\mathcal {A}(A,B)\) is the arithmetic mean and \(\mathcal {G}(A,B)\) is one of the different versions of the geometric mean. When \(\mathcal {G}(A,B)=A^{1/2}B^{1/2}\) this distance is \(\Vert A^{1/2}-B^{1/2}\Vert _2,\) and when \(\mathcal {G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}\) it is the Bures–Wasserstein metric. We study two other cases: \(\mathcal {G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},\) the Pusz–Woronowicz geometric mean, and \(\mathcal {G}(A,B)=\exp \big (\frac{\log A+\log B}{2}\big ),\) the log Euclidean mean. With these choices, d(AB) is no longer a metric, but it turns out that \(d^2(A,B)\) is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of m positive definite matrices with respect to these distance measures. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy.


Geometric mean Matrix divergence Bregman divergence Relative entropy Strict convexity Barycentre 

Mathematics Subject Classification

15B48 49K35 94A17 81P45 



The authors thank F. Hiai and S. Sra for helpful comments and references, and the anonymous referee for a careful reading of the manuscript. The first author is grateful to INRIA and Ecole polytechnique, Palaiseau for visits that facilitated this work, and to CSIR(India) for the award of a Bhatnagar Fellowship.


  1. 1.
    Abatzoglou, T.J.: Norm derivatives on spaces of operators. Math. Ann. 239, 129–135 (1979)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. SIAM J. Math. Anal. Appl. 43, 904–924 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aiken, J.G., Erdos, J.A., Goldstein, J.A.: Unitary approximation of positive operators. Illinois J. Math. 24, 61–72 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Amari, S.: Information Geometry and its Applications. Springer, Tokyo (2016)CrossRefGoogle Scholar
  5. 5.
    Ando, T.: Concavity of certain maps on positive definite matrices and applications to Hadamard products. Linear Algebra Appl. 26, 203–241 (1979)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ando, T., Li, C.-K., Mathias, R.: Geometric means. Linear Algebra Appl. 385, 305–334 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Math. Anal. Appl. 29, 328–347 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Banerjee, A., Merugu, S., Dhillon, I.S., Ghosh, J.: Clustering with Bregman divergences. J. Mach. Learn. Res. 6, 1705–1749 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Barbaresco, F.: Innovative tools for radar signal processing based on Cartan’s geometry of SPD matrices and information geometry. In: IEEE Radar Conference, Rome (2008)Google Scholar
  10. 10.
    Bauschke, H.H., Borwein, J.M.: Legendre functions and the method of random Bregman projections. J. Convex Anal. 4, 27–67 (1997)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bauschke, H.H., Borwein, J.M.: Joint and separate convexity of the Bregman distance. Stud. Comput. Math. 8, 23–36 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  13. 13.
    Bhagwat, K.V., Subramanian, R.: Inequalities between means of positive operators. Math. Proc. Camb. Philos. Soc. 83, 393–401 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bhatia, R.: Matrix Analysis. Springer, Tokyo (1997)CrossRefGoogle Scholar
  15. 15.
    Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)zbMATHGoogle Scholar
  16. 16.
    Bhatia, R.: The Riemannian mean of positive matrices. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 35–51. Springer, Tokyo (2013)CrossRefGoogle Scholar
  17. 17.
    Bhatia, R., Grover, P.: Norm inequalities related to the matrix geometric mean. Linear Algebra Appl. 437, 726–733 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bhatia, R., Jain, T., Lim, Y.: On the Bures-Wasserstein distance between positive definite matrices. Expos. Math. (2018).
  19. 19.
    Bhatia, R., Jain, T., Lim, Y.: Strong convexity of sandwiched entropies and related optimization problems. Rev. Math. Phys. 30, 1850014 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Carlen, E.A., Lieb, E.H.: A Minkowski type trace inequality and strong subadditivity of quantum entropy. Adv. Math. Sci. AMS Transl. 180, 59–68 (1999)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Carlen, E.A., Lieb, E.H.: A Minkowski type trace inequality and strong subadditivity of quantum entropy. II. Convexity and concavity. Lett. Math. Phys. 83, 107–126 (2008)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Chebbi, Z., Moakher, M.: Means of Hermitian positive-definite matrices based on the log-determinant \(\alpha \)-divergence function. Linear Algebra Appl. 436, 18721889 (2012)Google Scholar
  23. 23.
    Dhillon, I.S., Tropp, J.A.: Matrix nearness problems with Bregman divergences. SIAM J. Matrix Anal. Appl. 29, 1120–1146 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Fletcher, P., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87, 250–262 (2007)CrossRefGoogle Scholar
  25. 25.
    Hiai, F., Mosonyi, M., Petz, D., Beny, C.: Quantum f-divergences and error correction. Rev. Math. Phys. 23, 691–747 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Jencová, A.: Geodesic distances on density matrices. J. Math. Phys. 45, 1787–1794 (2004)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Jencova, A., Ruskai, M.B.: A unified treatment of convexity of relative entropy and related trace functions with conditions for equality. Rev. Math. Phys. 22, 1099–1121 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lim, Y., Palfia, M.: Matrix power means and the Karcher mean. J. Funct. Anal. 262, 1498–1514 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Modin, K.: Geometry of matrix decompositions seen through optimal transport and information geometry. J. Geom. Mech. 9, 335–390 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Nielsen, F., Bhatia, R. (eds.): Matrix Information Geometry. Springer, Tokyo (2013)Google Scholar
  31. 31.
    Nielsen, F., Boltz, S.: The Burbea-Rao and Bhattacharyya centroids. IEEE Trans. Inf. Theory 57, 5455–5466 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Pitrik, J., Virosztek, D.: On the joint convexity of the Bregman divergence of matrices. Lett. Math. Phys. 105, 675–692 (2015)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Pusz, W., Woronowicz, S.L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8, 159–170 (1975)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefGoogle Scholar
  35. 35.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Tokyo (1998)CrossRefGoogle Scholar
  36. 36.
    Sra, S.: Positive definite matrices and the \(S\)-divergence. Proc. Am. Math. Soc. 144, 2787–2797 (2016)Google Scholar
  37. 37.
    Takatsu, A.: Wasserstein geometry of Gaussian measures. Osaka J. Math. 48, 1005–1026 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Ashoka UniversitySonepatIndia
  2. 2.INRIA and CMAPEcole Polytechnique, CNRSPalaiseauFrance
  3. 3.Indian Statistical InstituteNew DelhiIndia

Personalised recommendations