The Fisher-Rao distance between two probability distributions of a
statistical model is defined as the Riemannian geodesic distance induced by the
Fisher information metric. In order to calculate the Fisher-Rao distance in
closed-form, we need (1) to elicit a formula for the Fisher-Rao geodesics, and
(2) to integrate the Fisher length element along those geodesics. We consider
several numerically robust approximation and bounding techniques for the
Fisher-Rao distances: First, we report generic upper bounds on Fisher-Rao
distances based on closed-form 1D Fisher-Rao distances of submodels. Second, we
describe several generic approximation schemes depending on whether the
Fisher-Rao geodesics or pregeodesics are available in closed-form or not. In
particular, we obtain a generic method to guarantee an arbitrarily small
additive error on the approximation provided that Fisher-Rao pregeodesics and
tight lower and upper bounds are available. Third, we consider the case of
Fisher metrics being Hessian metrics, and report generic tight upper bounds on
the Fisher-Rao distances using techniques of information geometry.
Uniparametric and biparametric statistical models always have Fisher Hessian
metrics, and in general a simple test allows to check whether the Fisher
information matrix yields a Hessian metric or not. Fourth, we consider
elliptical distribution families and show how to apply the above techniques to
these models. We also propose two new distances based either on the Fisher-Rao
lengths of curves serving as proxies of Fisher-Rao geodesics, or based on the
Birkhoff/Hilbert projective cone distance. Last, we consider an alternative
group-theoretic approach for statistical transformation models based on the
notion of maximal invariant which yields insights on the structures of the
Fisher-Rao distance formula which may be used fruitfully in applications.

Fisher-Rao 距离是由 Fisher 信息度量引起的 Riemannian 测地距离，本文通过计算 Fisher-Rao 距离的极限形式、数值逼近和上下界等方法来研究和应用该距离。

Fisher-Rao 距离的近似和界定技术

Approximation and bounding techniques for the Fisher-Rao distances

Data sets of multivariate normal distributions abound in many scientific
areas like diffusion tensor imaging, structure tensor computer vision, radar
signal processing, machine learning, just to name a few. In order to process
those normal data sets for downstream tasks like filtering, classification or
clustering, one needs to define proper notions of dissimilarities between
normals and paths joining them. The Fisher-Rao distance defined as the
Riemannian geodesic distance induced by the Fisher information metric is such a
principled metric distance which however is not known in closed-form excepts
for a few particular cases. In this work, we first report a fast and robust
method to approximate arbitrarily finely the Fisher-Rao distance between
multivariate normal distributions. Second, we introduce a class of distances
based on diffeomorphic embeddings of the normal manifold into a submanifold of
the higher-dimensional symmetric positive-definite cone corresponding to the
manifold of centered normal distributions. We show that the projective Hilbert
distance on the cone yields a metric on the embedded normal submanifold and we
pullback that cone distance with its associated straight line Hilbert cone
geodesics to obtain a distance and smooth paths between normal distributions.
Compared to the Fisher-Rao distance approximation, the pullback Hilbert cone
distance is computationally light since it requires to compute only the extreme
minimal and maximal eigenvalues of matrices. Finally, we show how to use those
distances in clustering tasks.

本文介绍了一种快速且鲁棒的方法来近似计算多变量正态分布之间的 Fisher-Rao 距离，同时将正态分布流形通过微分同胚嵌入到高维对称正定锥面的子流形中，并使用锥面上的投影 Hilbert 距离作为距离度量，通过拉回锥面距离来获得正态分布之间的距离和平滑路径，这种方法在计算复杂度上比 Fisher-Rao 距离近似方法更轻量级，在聚类任务中具有应用价值。

多元正态分布之间的 Fisher-Rao 距离和拉回 SPD 圆锥距离

Fisher-Rao distance and pullback SPD cone distances between multivariate  normal distributions

Although data diffusion embeddings are ubiquitous in unsupervised learning
and have proven to be a viable technique for uncovering the underlying
intrinsic geometry of data, diffusion embeddings are inherently limited due to
their discrete nature. To this end, we propose neural FIM, a method for
computing the Fisher information metric (FIM) from point cloud data - allowing
for a continuous manifold model for the data. Neural FIM creates an extensible
metric space from discrete point cloud data such that information from the
metric can inform us of manifold characteristics such as volume and geodesics.
We demonstrate Neural FIM's utility in selecting parameters for the PHATE
visualization method as well as its ability to obtain information pertaining to
local volume illuminating branching points and cluster centers embeddings of a
toy dataset and two single-cell datasets of IPSC reprogramming and PBMCs
(immune cells).

提出一种名为神经 FIM 的新方法，可以从离散点云数据中计算 Fisher 信息度量，并创建一个可扩展的度量空间，以建立连续流形模型来生成嵌入数据，以量化该流形的各种特征，证明了神经 FIM 在选择 PHATE 可视化方法的参数方面的实用性以及在 IPSC 重编程和 PBMCs（免疫细胞）等单细胞数据集的分支点和聚类中心嵌入的局部体积方面获取的信息。