# Stochastic Development Regression on Non-Linear Manifolds

## TL;DR

This paper presents a novel regression model for data on non-linear manifolds using stochastic development of Euclidean processes, enabling analysis of complex geometric data like anatomical shapes.

## Contribution

It introduces a stochastic development-based regression framework for manifold-valued data, incorporating geometric structure and maximum likelihood estimation with Laplace approximation.

## Key findings

- Model effectively captures relationships between manifold data and Euclidean variables.
- Simulation studies demonstrate accurate parameter estimation.
- Application to Corpus Callosum shapes shows practical utility.

## Abstract

We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is based on stochastic development of Euclidean diffusion processes to the manifold. Defining the data distribution as the transition distribution of the mapped stochastic process, parameters of the model, the non-linear analogue of design matrix and intercept, are found via maximum likelihood. The model is intrinsically related to the geometry encoded in the connection of the manifold. We propose an estimation procedure which applies the Laplace approximation of the likelihood function. A simulation study of the performance of the model is performed and the model is applied to a real dataset of Corpus Callosum shapes.