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.

翻译：本文针对三角可剖分流形上的回归问题，发展了非平衡哈尔小波树集成方法。给定三角剖分流形上的采样数据，我们构建了原子支撑于测地三角形上的非平衡哈尔小波树，其在$L^2(μ_n)$空间中构成标准正交系（其中$μ_n$为样本经验测度），从而可将非平衡哈尔树作为加法集成中的弱学习器。我们的构造将经典非平衡哈尔小波树从规则欧几里得网格推广至一般三角可剖分流形，同时保留了三个关键特性：（一）在采样点处的正交性与精确重构性；（二）通过测地三角剖分实现适应流形几何结构的数据驱动递归划分；（三）与基于优化的集成构建及贝叶斯集成构建的兼容性。在欧几里得情形下，该框架退化为标准非平衡哈尔小波树回归，为比较提供了基准。我们通过在球面上的合成回归实验及球面网格上的气候异常场分析验证了该方法，其中三角剖分流形上的非平衡哈尔集成显著优于经典树集成与非自适应网格小波方法。为完备起见，我们还报告了规则网格上图像去噪的实验结果。贝叶斯变体方法（RUHWT）可为流形上的函数估计提供后验不确定性量化。代码实现已发布于http://www.github.com/hrluo/WaveletTrees。