The correlation dimension of natural language is measured by applying the
Grassberger-Procaccia algorithm to high-dimensional sequences produced by a
large-scale language model. This method, previously studied only in a Euclidean
space, is reformulated in a statistical manifold via the Fisher-Rao distance.
Language exhibits a multifractal, with global self-similarity and a universal
dimension around 6.5, which is smaller than those of simple discrete random
sequences and larger than that of a Barab\'asi-Albert process. Long memory is
the key to producing self-similarity. Our method is applicable to any
probabilistic model of real-world discrete sequences, and we show an
application to music data.

使用 Grassberger-Procaccia 算法和 Fisher-Rao 距离，测量了自然语言的相关维数，并证明了语言具有多分形性质，具有全局自相似性和普遍维数约为 6.5，大于简单离散随机序列的维数但小于 Barabasi-Albert 进程的维数，同时长时记忆是产生自相似性的关键。这种方法适用于任何真实世界离散序列的概率模型，并展示了音乐数据的应用。

在统计多概况中的自然语言相关维度

Correlation Dimension of Natural Language in a Statistical Manifold

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。