  Inspired by logistic regression, we introduce a regression model for data
tuples consisting of a binary response and a set of covariates residing in a
metric space without vector structures. Based on the proposed model we also
develop a binary classifier for metric-space valued data. We propose a maximum
likelihood estimator for the metric-space valued 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. To the best of our knowledge, the
proposed regression model and the above minimax bounds are the first of their
kind for analyzing a binary response with covariates residing in general metric
spaces. We also 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.
