Current tools for multivariate density estimation struggle when the density is concentrated near a non‐linear subspace or manifold. Most approaches require the choice of a kernel, with the multivariate Gaussian kernel by far the most commonly used. Although heavy‐tailed and skewed extensions have been proposed, such kernels cannot capture curvature in the support of the data. This leads to poor performance unless the sample size is very large relative to the dimension of the data. The paper proposes a novel generalization of the Gaussian distribution, which includes an additional curvature parameter. We refer to the proposed class as Fisher–Gaussian kernels, since they arise by sampling from a von Mises–Fisher density on the sphere and adding Gaussian noise. The Fisher–Gaussian density has an analytic form and is amenable to straightforward implementation within Bayesian mixture models by using Markov chain Monte Carlo sampling. We provide theory on large support and illustrate gains relative to competitors in simulated and real data applications.

中文翻译：

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使用 Fisher-Gaussian 核估计具有非线性支持的密度

当密度集中在非线性子空间或流形附近时，当前用于多元密度估计的工具会遇到困难。大多数方法都需要选择核，目前最常用的是多元高斯核。尽管已经提出了重尾和倾斜扩展，但这样的内核无法捕获数据支持中的曲率。这会导致性能不佳，除非样本量相对于数据维度非常大。本文提出了一种新的高斯分布推广，其中包括一个额外的曲率参数。我们将提出的类称为 Fisher-Gaussian 核，因为它们是通过从球体上的 von Mises-Fisher 密度采样并添加高斯噪声而产生的。Fisher-Gaussian 密度具有解析形式，并且可以通过使用马尔可夫链蒙特卡罗采样在贝叶斯混合模型中直接实现。我们提供有关大支持的理论，并说明在模拟和真实数据应用程序中相对于竞争对手的收益。