Geodesic Learning via Unsupervised Decision Forests

TL;DR

This paper introduces URerF, an unsupervised decision forest method that accurately learns geodesic distances in noisy high-dimensional data, outperforming existing manifold learning algorithms.

Contribution

The paper presents URerF, a novel unsupervised random forest approach that efficiently approximates geodesic distances in noisy, high-dimensional manifolds, with a new BIC-based split criterion.

Findings

URerF is robust to high-dimensional noise.

URerF outperforms Isomap, UMAP, and FLANN in geodesic distance estimation.

URerF accurately estimates geodesic distances on real connectome data.

Abstract

Geodesic distance is the shortest path between two points in a Riemannian manifold. Manifold learning algorithms, such as Isomap, seek to learn a manifold that preserves geodesic distances. However, such methods operate on the ambient dimensionality, and are therefore fragile to noise dimensions. We developed an unsupervised random forest method (URerF) to approximately learn geodesic distances in linear and nonlinear manifolds with noise. URerF operates on low-dimensional sparse linear combinations of features, rather than the full observed dimensionality. To choose the optimal split in a computationally efficient fashion, we developed a fast Bayesian Information Criterion statistic for Gaussian mixture models. We introduce geodesic precision-recall curves which quantify performance relative to the true latent manifold. Empirical results on simulated and real data demonstrate that URerF is robust to high-dimensional noise, where as other methods, such as Isomap, UMAP, and FLANN, quickly deteriorate in such settings. In particular, URerF is able to estimate geodesic distances on a real connectome dataset better than other approaches.