Mutual information quantifies the dependence between two random variables and
remains invariant under diffeomorphisms. In this paper, we explore the
pointwise mutual information profile, an extension of mutual information that
maintains this invariance. We analytically describe the profiles of
multivariate normal distributions and introduce the family of fine
distributions, for which the profile can be accurately approximated using Monte
Carlo methods. We then show how fine distributions can be used to study the
limitations of existing mutual information estimators, investigate the behavior
of neural critics used in variational estimators, and understand the effect of
experimental outliers on mutual information estimation. Finally, we show how
fine distributions can be used to obtain model-based Bayesian estimates of
mutual information, suitable for problems with available domain expertise in
which uncertainty quantification is necessary.

本文研究相互信息、逐点相互信息轮廓、多元正态分布、蒙特卡洛方法，该研究还使用精细分布来研究现有相互信息估计器的局限性、变分估计器中使用的神经评论家的行为以及实验异常值对相互信息估计的影响，并展示了如何使用精细分布来获得基于模型的贝叶斯估计的相互信息，适用于具有领域专业知识且需要不确定性量化的问题。

混合物和神经评论家：关于细度分布的点互信息特征

The Mixtures and the Neural Critics: On the Pointwise Mutual Information  Profiles of Fine Distributions

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

We consider the hypothesis testing problem of detecting a shift between the
means of two multivariate normal distributions in the high-dimensional setting,
allowing for the data dimension p to exceed the sample size n. Specifically, we
propose a new test statistic for the two-sample test of means that integrates a
random projection with the classical Hotelling T^2 statistic. Working under a
high-dimensional framework with (p,n) tending to infinity, we first derive an
asymptotic power function for our test, and then provide sufficient conditions
for it to achieve greater power than other state-of-the-art tests. Using ROC
curves generated from synthetic data, we demonstrate superior performance
against competing tests in the parameter regimes anticipated by our theoretical
results. Lastly, we illustrate an advantage of our procedure's false positive
rate with comparisons on high-dimensional gene expression data involving the
discrimination of different types of cancer.

在高维情形下考虑了检验两个多元正态分布均值差异的统计假设检验问题，其中引入了投影方法和 Hotelling T^2 统计量，并针对高维条件下的渐近推理，概述了测试的渐近功效函数以及通往提高其他最先进测试功效的充分条件，最后通过基于 ROC 曲线的实验，验证了该检验方法在高维数据中区分肿瘤数据种类时的优良性能。