非中心缩放高斯分布混合的黎曼优化
Riemannian optimization for non-centered mixture of scaled Gaussian distributions
AI总结：
本文提出基于Fisher-Rao信息几何的黎曼梯度下降算法，用于非中心缩放高斯分布混合的正则化负对数似然最小化和质心计算，并利用KL散度实现最近质心分类器，在Breizhcrops数据集上验证了性能与鲁棒性。
摘要：
本文研究了非中心缩放高斯分布混合（NC-MSG）的统计模型。利用与该分布相关的Fisher-Rao信息几何，我们推导出一种黎曼梯度下降算法。该算法被用于解决两个最小化问题。第一个是正则化负对数似然（NLL）的最小化，后者在白色高斯分布与NC-MSG之间进行权衡。给出了正则化的条件，使得在没有对样本做假设的情况下，该问题的最小值存在性得到保证。然后，推导了两个NC-MSG之间的Kullback-Leibler（KL）散度。该散度使我们能够定义一个最小化问题来计算多个NC-MSG的质心。所提出的黎曼梯度下降算法被用于解决这第二个最小化问题。数值实验表明，黎曼梯度下降在这两个问题上具有良好的性能和速度。最后，利用KL散度及其相关的质心实现了一个最近质心分类器。应用于大规模数据集Breizhcrops，该分类器显示出良好的准确性以及对测试集刚性变换的鲁棒性。
英文摘要：
This paper studies the statistical model of the non-centered mixture of scaled Gaussian distributions (NC-MSG). Using the Fisher-Rao information geometry associated to this distribution, we derive a Riemannian gradient descent algorithm. This algorithm is leveraged for two minimization problems. The first one is the minimization of a regularized negative log-likelihood (NLL). The latter makes the trade-off between a white Gaussian distribution and the NC-MSG. Conditions on the regularization are given so that the existence of a minimum to this problem is guaranteed without assumptions on the samples. Then, the Kullback-Leibler (KL) divergence between two NC-MSG is derived. This divergence enables us to define a minimization problem to compute centers of mass of several NC-MSGs. The proposed Riemannian gradient descent algorithm is leveraged to solve this second minimization problem. Numerical experiments show the good performance and the speed of the Riemannian gradient descent on the two problems. Finally, a Nearest centroid classifier is implemented leveraging the KL divergence and its associated center of mass. Applied on the large scale dataset Breizhcrops, this classifier shows good accuracies as well as robustness to rigid transformations of the test set.
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