Wavelet Tree Ensembles for Triangulable Manifolds

TL;DR

The paper develops orthogonal unbalanced Haar wavelet trees (UHWT) for regression on triangulable manifolds by coupling data-driven geodesic triangulations with an orthonormal $L^2(\mu_n)$ basis. It then builds additive ensembles via boosting and Bayesian backfitting (RUHWT) to achieve accurate predictions and quantify uncertainty, extending UHWT from Euclidean grids to manifolds. The authors prove oracle-type bounds that favor sparse UH representations and demonstrate superior performance on simulated sphere data and climate data, compared to classical tree ensembles and fixed-mesh wavelets. The work provides a scalable, geometry-aware regression framework for scalar-on-manifold problems with practical implementations and publicly available code.

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 $L^2(μ_n)$, where $μ_n$ 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.

Figures (13)

• Figure 1: Comparison between the Balanced Haar (BH) and UH optimization approach on a noisy image that contains a star and a mostly two tone background. The first row uses BH splits. The second row uses UH splits chosen by optimization; these splits delineate the boundary of the star and adapt to the volume of the main object. At depth 8 the partitions concentrate along the edges of the star. The final reconstruction with a single tree preserves both the boundary and the volume of the star while reducing background interference.
• Figure 2: First panel: MAP partition overlaid on the input image. Posterior mean (second panel), standard deviation (third panel), and 95% credible interval width (fourth panel) computed from 500 full posterior draws of a Bayesian additive UH-tree model with 200 trees and geometric early-stopping prior probability $0.5^{\text{node depth}}$.
• Figure 3: Examples of triangulable manifolds and four approaches (balance, balance4, adapt, adapt_vertex) to subdivide a triangle. Points represent data; the red point indicates the pivot point, and dashed lines show the cutting edge. The balance and balance4 schemes do not depend on data. The adapt scheme uses data to choose which edge to bisect at its midpoint. The adapt_vertex scheme uses data to choose both the edge and the cut point.
• Figure 4: Metrics of methods fit to a $512\times512$ pixel astronaut image with additive Gaussian noise whose standard deviation equals the noiseless image's standard deviation. For each boosted ensemble, the learning rate is 0.1. The hyperparameter $b$ is as defined at the beginning of Sec \ref{['subsec:numerical-image']}. For the fourth panel, the test MSEs for $b\in\{0.08,0.1\}$ were larger than $0.018$ and hence are not shown.
• Figure 5: Empirical comparison of tree ensemble models (“ RR” refers to the method version that randomly rotates covariates before fitting a base learner) trained on $n=1941$ points $(\bm{x},y)$ from the GISS dataset. Trees are allowed to grow to full depth. Boosted methods use learning rate $0.05$. Top two rows: predictions of the ensemble methods. The 7766 test points are shown in the top-right sphere. Third row: MSE on 7766 test points.

Theorems & Definitions (18)

• Theorem 2.1
• Theorem 5.1: Oracle bounds on a fixed partition
• Corollary 5.2: Sparse detail vectors
• Lemma B.1: Orthonormality of the UH system
• proof
• Theorem C.1
• proof
• Theorem E.1
• proof
• Theorem G.1