为对称正定矩阵流形设计了新型非各向同性包裹高斯分布。
Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices
- 基于指数映射构建流形上的非各向同性包裹高斯模型。
- 在合成与真实数据集上验证了该分布的鲁棒性和灵活性。
- 适合研究流形数据建模与几何感知机器学习的学者。
循环和非平坦的数据分布广泛存在于数据科学的多个领域，但其特定的几何结构在机器学习框架中常被忽视。充分考虑这些数据的底层几何特性至关重要，尤其是在扩展如广泛应用的高斯分布等统计模型时。本文聚焦于信息几何中的对称正定（SPD）矩阵流形，提出一种基于指数映射的非各向同性包裹高斯分布，推导了其理论性质，并提出了最大似然参数估计方法。此外，我们从概率视角重新诠释了现有SPD分类器，并基于包裹高斯模型提出了新的分类方法。在合成与真实数据集上的实验表明，该几何感知分布具有良好的鲁棒性和灵活性，凸显其在推进基于流形的数据分析中的潜力。本工作为将经典机器学习与统计方法推广至更复杂结构化数据奠定了基础。
原文摘要 · Abstract (English)
Circular and non-flat data distributions are prevalent across diverse domains of data science, yet their specific geometric structures often remain underutilized in machine learning frameworks. A principled approach to accounting for the underlying geometry of such data is pivotal, particularly when extending statistical models, like the pervasive Gaussian distribution. In this work, we tackle those issue by focusing on the manifold of symmetric positive definite (SPD) matrices, a key focus in information geometry. We introduce a non-isotropic wrapped Gaussian by leveraging the exponential map, we derive theoretical properties of this distribution and propose a maximum likelihood framework for parameter estimation. Furthermore, we reinterpret established classifiers on SPD through a probabilistic lens and introduce new classifiers based on the wrapped Gaussian model. Experiments on synthetic and real-world datasets demonstrate the robustness and flexibility of this geometry-aware distribution, underscoring its potential to advance manifold-based data analysis. This work lays the groundwork for extending classical machine learning and statistical methods to more complex and structured data.
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