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Matrix versions of the Hellinger distance

Published in Springer

On the space of positive definite matrices, we consider distance functions of the form 𝑑(𝐴,𝐡)=[trπ’œ(𝐴,𝐡)βˆ’tr𝒒(𝐴,𝐡)]1/2,d(A,B)=[trA(A,B)βˆ’trG(A,B)]1/2, where π’œ(𝐴,𝐡)A(A,B) is the arithmetic mean and 𝒒(𝐴,𝐡)G(A,B) is one of the different versions of the geometric mean. When 𝒒(𝐴,𝐡)=𝐴1/2𝐡1/2G(A,B)=A1/2B1/2 this distance is ‖𝐴1/2βˆ’π΅1/2β€–2,β€–A1/2βˆ’B1/2β€–2, and when 𝒒(𝐴,𝐡)=(𝐴1/2𝐡𝐴1/2)1/2G(A,B)=(A1/2BA1/2)1/2 it is the Bures–Wasserstein metric. We study two other cases: 𝒒(𝐴,𝐡)=𝐴1/2(π΄βˆ’1/2π΅π΄βˆ’1/2)1/2𝐴1/2,G(A,B)=A1/2(Aβˆ’1/2BAβˆ’1/2)1/2A1/2, the Pusz–Woronowicz geometric mean, and 𝒒(𝐴,𝐡)=exp(log𝐴+log𝐡2),G(A,B)=exp⁑(log⁑A+log⁑B2), the log Euclidean mean. With these choices, d(A, B) is no longer a metric, but it turns out that 𝑑2(𝐴,𝐡)d2(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.

About the journal
JournalData powered by TypesetLetters in Mathematical Physics
PublisherData powered by TypesetSpringer
Open AccessNo