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Register a new userUniqueness of the Fisher-Rao metric on the space of smooth densities
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On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher--Rao metric.
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It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form $$ G_mu(alpha,beta)=C_1(mu(M)) int_M frac{alpha}{mu}frac{beta}{mu},mu + C_2(mu(M)) int_Malpha cdot int_Mbeta $$ for some smooth functions $C_1,C_2$ of the total volume $mu(M)$. Here we determine the geodesics and the curvature of this metric and study geodesic and metric completeness.
In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is geodesically complete and has everywhere negative sectional curvature. An important consequence of this negative curvature for applications is that the Fr{e}chet mean of a set of Dirichlet distributions is uniquely defined in this geometry.
In our previous work [Chen el al., J. Comput. Phys., 373(2018)], the quadratic Wasserstein metric is successfully applied to the earthquake location problem. The actual earthquake hypocenter can be accurately recovered starting from initial values very far from the true ones. However, the seismic wave signals need to be normalized since the quadratic Wasserstein metric requires mass conservation. This brings a critical difficulty. Since the amplitude of a seismogram at a receiver is a good representation of the distance between the source and the receiver, simply normalizing the signals will cause the objective function in optimization process to be insensitive to the distance between the source and the receiver. When the data is contaminated with strong noise, the minimum point of the objective function will deviate and lead to a low accurate location result. To overcome the difficulty mentioned above, we apply the Wasserstein-Fisher-Rao (WFR) metric [Chizat et al., Found. Comput. Math., 18(2018)] to the earthquake location problem. The WFR metric is one of the newly developed metric in the unbalanced Optimal Transport theory. It does not require the normalization of the seismic signals. Thus, the amplitude of seismograms can be considered as a new constraint, which can substantially improve the sensitivity of the objective function to the distance between the source and the receiver. As a result, we can expect more accurate location results from the WFR metric based method compare to those based on quadratic Wasserstein metric under high-intensity noise. The numerical examples also demonstrate this.
This paper focuses on the study of open curves in a manifold M, and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M = Imm([0,1], M) by pullback of a metric on the tangent bundle TM derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on TM induces a first-order Sobolev metric on M with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the SRV representations of the curves. The geodesic equations for this metric are given, as well as an idea of how to compute the exponential map for observed trajectories in applications. This provides a generalized theoretical SRV framework for curves lying in a general manifold M .
The purpose of this article is to exploit the geometric structure of Quantum Mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannons Entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.
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