Wavelet Tree Ensembles for Triangulable Manifolds

2026-01-28

AI Summary

This work addresses the challenge of balancing geometric adaptivity and reconstruction accuracy in regression modeling on triangulable manifolds. We propose an ensemble model based on geodesic triangles using unbalanced Haar wavelet trees. Our method extends unbalanced Haar wavelets—previously limited to Euclidean grids—to general manifolds for the first time, recursively partitioning the manifold in a data-driven manner to capture its intrinsic geometry. These wavelet trees serve as weak learners within an additive ensemble framework, offering orthogonality, exact reconstruction capability, and seamless compatibility with Bayesian and optimization-based inference schemes. Experiments demonstrate that the proposed model significantly outperforms classical tree ensembles and non-adaptive wavelet approaches on tasks including synthetic regression on the sphere and climate anomaly field prediction, while also showing strong performance in image denoising.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• regression
• triangulable manifolds
• geodesic triangulations
• wavelet trees
• manifold learning

Innovation

• Unbalanced Haar wavelets
• Triangulable manifolds
• Geodesic triangulation
• Wavelet tree ensembles
• Bayesian uncertainty quantification