Wavelet Tree Ensembles for Triangulable Manifolds

TL;DR

This paper introduces unbalanced Haar wavelet tree ensembles tailored for regression on triangulable manifolds, extending classical wavelet methods to complex geometries and demonstrating superior performance on synthetic and real-world data.

Contribution

It develops a novel wavelet tree ensemble framework for manifolds, maintaining orthogonality, adaptive partitioning, and compatibility with ensemble methods, extending Euclidean wavelets to complex geometries.

Findings

Outperforms classical tree ensembles on spherical and climate data

Provides exact reconstruction and adaptive partitioning on manifolds

Includes a Bayesian variant for uncertainty quantification

Abstract

We develop unbalanced Haar (UH) wavelet tree ensembles for regression on triangulable manifolds. Given data sampled on a triangulated manifold, we construct UH wavelet trees whose atoms are supported on geodesic triangles and form an orthonormal system in $L^2(\mu_n)$, where $\mu_n$ is the empirical measure on the sample, which allows us to use UH trees as weak learners in additive ensembles. Our construction extends classical UH wavelet trees from regular Euclidean grids to generic triangulable manifolds while preserving three key properties: (i) orthogonality and exact reconstruction at the sampled locations, (ii) recursive, data-driven partitions adapted to the geometry of the manifold via geodesic triangulations, and (iii) compatibility with optimization-based and Bayesian ensemble building. In Euclidean settings, the framework reduces to standard UH wavelet tree regression and provides a baseline for comparison. We illustrate the method on synthetic regression on the sphere and on climate anomaly fields on a spherical mesh, where UH ensembles on triangulated manifolds substantially outperform classical tree ensembles and non-adaptive mesh-based wavelets. For completeness, we also report results on image denoising on regular grids. A Bayesian variant (RUHWT) provides posterior uncertainty quantification for function estimates on manifolds. Our implementation is available at http://www.github.com/hrluo/WaveletTrees.