# Binary Regression and Classification with Covariates in Metric Spaces

## TL;DR

This paper introduces a novel regression and classification framework for binary data with covariates in general metric spaces, providing theoretical bounds and demonstrating practical effectiveness through simulations and fMRI data analysis.

## Contribution

It develops the first regression model and minimax bounds for binary responses with covariates in general metric spaces, including Riemannian manifolds.

## Key findings

- Proposed estimator achieves optimal error bounds in key metric spaces.
- Classifier's excess risk is bounded and shown to be optimal in Riemannian manifolds.
- Numerical experiments confirm the practical utility of the methods.

## Abstract

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.