We introduce a logistic regression model for data pairs consisting of a binary response and a covariate residing in a non-Euclidean metric space without vector structures. Based on the proposed model we also develop a binary classifier for non-Euclidean objects. We propose a maximum likelihood estimator for the non-Euclidean regression coefficient in the model, and provide upper bounds on the estimation error under various metric entropy conditions that quantify complexity of the underlying metric space. Matching lower bounds are derived for the important metric spaces commonly seen in statistics, establishing optimality of the proposed estimator in such spaces. Similarly, an upper bound on the excess risk of the developed classifier is provided for general metric spaces. A finer upper bound and a matching lower bound, and thus optimality of the proposed classifier, are established for Riemannian manifolds. We investigate the numerical performance of the proposed estimator and classifier via simulation studies, and illustrate their practical merits via an application to task-related fMRI data.
翻译:我们对数据配对采用一种后勤回归模型,由二进制反应和居住在无矢量结构的非欧洲度空间的共变体组成。根据拟议的模型,我们还为非欧洲的天体开发了一个二进制分类器。我们为模型中的非欧洲的回归系数提出了一个最大可能性估计值,并为各种能量化基本计量空间复杂性的参数性能条件下的估计误差提供了上限。为统计中常见的重要指标空间设定了匹配的下限,为这些空间中拟议的估计器建立了最佳性能。同样,为一般的公制空间提供了对开发的分类器超重风险的上限。为里曼河道设置了一个细框和匹配的下限,从而为拟议的分类器的最佳性能。我们通过模拟研究,调查拟议的估计器和分类器的数值性能,并通过应用与任务相关的FMRI数据来说明其实际优点。