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Correction to: Matrix versions of the Hellinger distance

  • Rajendra Bhatia
  • Stephane Gaubert
  • Tanvi JainEmail author
Correction
  • 73 Downloads

1 Correction to: Letters in Mathematical Physics (2019) 109:1777–1804  https://doi.org/10.1007/s11005-019-01156-0

Theorem 9 in our paper [1] is wrong. The statement should be replaced by the following:

Theorem

When d = d3, the minimum in (13) is attained at a unique point X which is the solution of the matrix equation
$$ X^{2}\,=\,\frac{2}{\pi }\sum\limits_{j\,=\,1}^{m} {w_{j} } \int\limits_{0}^{\infty } {\left( {\lambda X^{ - 1} + A_{j}^{ - 1} } \right)^{ - 2} \sqrt \lambda d\lambda}. $$
(1)
This minimiser is the 1/2-power mean Q1/2given by (14) if Q1/2commutes with every Aj. In particular, the minimiser is Q1/2if
  1. (i)

    All A j ’s commute, or

     
  2. (ii)

    Q1/2 =  I.

     

An outline of Proof

Let
$$ f(X)\,=\,\sum\limits_{j\,=\,1}^{n} {w_{j} \Phi_{3} (X,A_{j} )} $$
be the objective function (13) of [1]. Using Proposition 1 we see that
$$ \begin{aligned} Df(X)(Y) &\,=\,{\text{tr}}\left( {Y - 2\sum\limits_{j\,=\,1}^{m} {w_{j} } \int\limits_{0}^{\infty } {\left( {\lambda + XA_{j}^{ - 1} } \right)^{ - 1} Y\left( {\lambda + A_{j}^{ - 1} X} \right)^{ - 1} {\text{d}}\nu (\lambda )} } \right) \\ & = {\text{tr}}\left( {\left( {I - 2\sum\limits_{j\,=\,1}^{m} {w_{j} } \int\limits_{0}^{\infty } {\left( {\left( {\lambda + XA_{j}^{ - 1} } \right)\left( {\lambda + A_{j}^{ - 1} X} \right)} \right)^{ - 1} {\text{d}}\nu (\lambda )} } \right)Y} \right). \\ \end{aligned} $$
Hence X0 is a critical point of f if and only if X0 satisfies the matrix equation
$$ I = 2\sum\limits_{j\,=\,1}^{m} {w_{j} } \int\limits_{0}^{\infty } {\left( {\left( {\lambda + XA_{j}^{ - 1} } \right)\left( {\lambda + A_{j}^{ - 1} X} \right)} \right)^{ - 1} {\text{d}}\nu (\lambda )} . $$
(2)
Taking congruence with X on both sides, we see that this equation is equivalent to (1).

The function in (13) is strictly convex. So it has at most one critical point. This entails the uniqueness of the solution to (1). The existence of a solution follows from the Brouwer fixed point theorem. The argument is similar to that in our proof of Theorem 10.

On differentiating (25) of [1], we get
$$ \frac{1}{2}x^{ - 1/2} = \int\limits_{0}^{\infty } {\frac{1}{{(\lambda + x)^{2} }}{\text{d}}\nu (\lambda ).} $$
(3)
So if \( Q_{1/2} A_{j}^{ - 1} = A_{j}^{ - 1} Q_{1/2} \) for all \( 1\leqslant j \leqslant m\), from (2) and (3) we get
$$ \begin{aligned} I & = Q_{1/2}^{1/2} Q_{1/2}^{ - 1/2} = \sum\limits_{j\,=\,1}^{m} {w_{j} \left( {A_{j}^{1/2} Q_{1/2}^{ - 1/2} } \right)} \\ & = \sum\limits_{j\,=\,1}^{m} {w_{j} \left( {A_{j} Q_{1/2}^{ - 1} } \right)^{1/2} } \\ & = 2\sum\limits_{j\,=\,1}^{m} {w_{j} } \int\limits_{0}^{\infty } {(\lambda + A_{j}^{ - 1} Q_{1/2} )^{ - 2} {\text{d}}\nu (\lambda )} \\ & = 2\sum\limits_{j\,=\,1}^{m} {w_{j} } \int\limits_{0}^{\infty } {\left( {(\lambda + Q_{1/2} A_{j}^{ - 1} )(\lambda + A_{j}^{ - 1} Q_{1/2} )} \right)^{ - 1} {\text{d}}\nu (\lambda )} . \\ \end{aligned} $$
This proves the second statement of the theorem.□

We became aware of the mistake in Theorem 9 of [1] from the arxiv preprint [2]. The authors there give a correct version (in a different notation) and also give an example to show that the barycentre obtained in (1) does not reduce to formula (17) in our paper [1].

Notes

References

  1. 1.
    Bhatia, R., Gaubert, S., Jain, T.: Matrix versions of the Hellinger distance. Lett. Math. Phys. 109, 1777–1804 (2019)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Pitrik, J., Virosztek, D.: Quantum Hellinger distances revisited. arXiv:1903.10455v3

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

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

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