# Random Forest regression for manifold-valued responses

## TL;DR

This paper introduces a non-parametric Random Forest-based method for predicting complex manifold-valued data, addressing challenges in modeling non-linear geometric spaces common in biomedical and computer vision applications.

## Contribution

It develops a versatile distance-based Random Forest regression approach applicable to manifold-valued responses without requiring explicit manifold knowledge.

## Key findings

- Superior predictive performance in simulations
- Effective in image completion tasks
- Applicable even when manifold structure is unknown

## Abstract

An increasing array of biomedical and computer vision applications requires the predictive modeling of complex data, for example images and shapes. The main challenge when predicting such objects lies in the fact that they do not comply to the assumptions of Euclidean geometry. Rather, they occupy non-linear spaces, a.k.a. manifolds, where it is difficult to define concepts such as coordinates, vectors and expected values. In this work, we construct a non-parametric predictive methodology for manifold-valued objects, based on a distance modification of the Random Forest algorithm. Our method is versatile and can be applied both in cases where the response space is a well-defined manifold, but also when such knowledge is not available. Model fitting and prediction phases only require the definition of a suitable distance function for the observed responses. We validate our methodology using simulations and apply it on a series of illustrative image completion applications, showcasing superior predictive performance, compared to various established regression methods.