Geometry of the Qτ family and McCann displacement geodesics

Identify the geometric meaning of the family of minimizers {Qτ}_{τ>0} for the matrix potential Φτ and determine whether this family is related to the McCann displacement geodesic by a reparametrization or instead defines a distinct interpolation on the positive-definite cone.

Background

The paper introduces a one-parameter family Φτ(S)=tr(Q(S)τ+MQ(S){-τ}) whose unique minimizer satisfies Qτ=M{1/(2τ)}. At τ=1, the minimizer yields the Bures–Wasserstein distance between Gaussian measures.

For τ≠1, the trace of the minimizer produces a different quantity, so the geometric path traced by Qτ is not identified. Its possible relation to the McCann optimal-transport interpolation, or to α-geodesics, remains unresolved.

References

Whether the family {Qτ }τ >0of Remark 12.6 is related to it by a reparametrization, or whether Qτ instead traces out a distinct interpolation on PD(k) (e.g. related to the α-geodesics of §8), remains open.

— Information Geometry of Gradient Flows  (2608.21152 - Yoshizawa, 21 Aug 2026) in Remark 12.7 and Section 12.5 concluding open questions

Construct the Legendre–Fenchel dual of Φτ and determine under what orthogonality (projection) condition a Pythagorean-type identity d(S, S′′)2 = d(S, S′)2 + d(S′, S′′)2 holds, in the spirit of §10 but for the Bures–Wasserstein rather than the Bregman/KL geometry.

— Information Geometry of Gradient Flows  (2608.21152 - Yoshizawa, 21 Aug 2026) in Section 12 concluding open questions, item 2

Verify directly, for all τ > 0, that Φτis a unitarily invariant strictly plurisubharmonic function on the relevant GL(k, C)-orbit (rather than invoking the general theorem as a black box), and identify any τfor which strict plurisubharmonicity could fail.

— Information Geometry of Gradient Flows  (2608.21152 - Yoshizawa, 21 Aug 2026) in Section 12 concluding open questions, item 3

Analyze the convergence rate and initial-value dependence of the gradient flow dQ/dt = − grad Φτ (Q) numerically, tracking Sτ → S1 as τ varies, in the spirit of the convergence analysis of §12.11.

— Information Geometry of Gradient Flows  (2608.21152 - Yoshizawa, 21 Aug 2026) in Section 12 concluding open questions, item 5