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1. Spatial properties of Bayesian unsupervised trees

   Liu, Linxi; Ma, Li (September 2024, Proceedings of Machine Learning Research)

   Agrawal, Shipra; Roth, Aaron (Ed.)

   Tree-based methods are popular nonparametric tools for capturing spatial heterogeneity and making predictions in multivariate problems. In unsupervised learning, trees and their ensembles have also been applied to a wide range of statistical inference tasks, such as multi-resolution sketching of distributional variations, localization of high-density regions, and design of efficient data compression schemes. In this paper, we study the spatial adaptation property of Bayesian tree-based methods in the unsupervised setting, with a focus on the density estimation problem. We characterize spatial heterogeneity of the underlying density function by using anisotropic Besov spaces, region-wise anisotropic Besov spaces, and two novel function classes as their extensions. For two types of commonly used prior distributions on trees under the context of unsupervised learning—the optional P{ó}lya tree (Wong and Ma, 2010) and the Dirichlet prior (Lu et al., 2013)—we calculate posterior concentration rates when the density function exhibits different types of heterogeneity. In specific, we show that the posterior concentration rate for trees is near minimax over the anisotropic Besov space. The rate is adaptive in the sense that to achieve such a rate we do not need any prior knowledge of the parameters of the Besov space. 

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2. Convergence rates of a class of multivariate density estimation methods based on adaptive partitioning

   Linxi Liu; Danga Li; Wing Hung Wong (January 2023, Journal of machine learning research)

   Density estimation is a building block for many other statistical methods, such as classification, nonparametric testing, and data compression. In this paper, we focus on a nonparametric approach to multivariate density estimation, and study its asymptotic properties under both frequentist and Bayesian settings. The estimated density function is obtained by considering a sequence of approximating spaces to the space of densities. These spaces consist of piecewise constant density functions supported by binary partitions with increasing complexity. To obtain an estimate, the partition is learned by maximizing either the likelihood of the corresponding histogram on that partition, or the marginal posterior probability of the partition under a suitable prior. We analyze the convergence rate of the maximum likelihood estimator and the posterior concentration rate of the Bayesian estimator, and conclude that for a relatively rich class of density functions the rate does not directly depend on the dimension. We also show that the Bayesian method can adapt to the unknown smoothness of the density function. The method is applied to several specific function classes and explicit rates are obtained. These include spatially sparse functions, functions of bounded variation, and Holder continuous functions. We also introduce an ensemble approach, obtained by aggregating multiple density estimates fit under carefully designed perturbations, and show that for density functions lying in a Holder space (H^(1,β), 0 < β ≤ 1), the ensemble method can achieve minimax convergence rate up to a logarithmic term, while the corresponding rate of the density estimator based on a single partition is suboptimal for this function class. 

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3. Convergence Rates of a Class of Multivariate Density Estimation Methods Based on Adaptive Partitioning

   Liu, Linxi; Li, Dangna; Wong, Wing Hung (January 2023, Journal of machine learning research)

   Density estimation is a building block for many other statistical methods, such as classification, nonparametric testing, and data compression. In this paper, we focus on a non-parametric approach to multivariate density estimation, and study its asymptotic properties under both frequentist and Bayesian settings. The estimated density function is obtained by considering a sequence of approximating spaces to the space of densities. These spaces consist of piecewise constant density functions supported by binary partitions with increasing complexity. To obtain an estimate, the partition is learned by maximizing either the likelihood of the corresponding histogram on that partition, or the marginal posterior probability of the partition under a suitable prior. We analyze the convergence rate of the maximum likelihood estimator and the posterior concentration rate of the Bayesian estimator, and conclude that for a relatively rich class of density functions the rate does not directly depend on the dimension. We also show that the Bayesian method can adapt to the unknown smoothness of the density function. The method is applied to several specific function classes and explicit rates are obtained. These include spatially sparse functions, functions of bounded variation, and Holder continuous functions. We also introduce an ensemble approach, obtained by aggregating multiple density estimates fit under carefully designed perturbations, and show that for density functions lying in a Holder space (H^(1,β),0<β≤1), the ensemble method can achieve minimax convergence rate up to a logarithmic term, while the corresponding rate of the density estimator based on a single partition is suboptimal for this function class. 

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4. Bayesian Regression Tree Ensembles that Adapt to Smoothness and Sparsity

   https://doi.org/10.1111/rssb.12293  

   Linero, Antonio R.; Yang, Yun (September 2018, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Summary Ensembles of decision trees are a useful tool for obtaining flexible estimates of regression functions. Examples of these methods include gradient-boosted decision trees, random forests and Bayesian classification and regression trees. Two potential shortcomings of tree ensembles are their lack of smoothness and their vulnerability to the curse of dimensionality. We show that these issues can be overcome by instead considering sparsity inducing soft decision trees in which the decisions are treated as probabilistic. We implement this in the context of the Bayesian additive regression trees framework and illustrate its promising performance through testing on benchmark data sets. We provide strong theoretical support for our methodology by showing that the posterior distribution concentrates at the minimax rate (up to a logarithmic factor) for sparse functions and functions with additive structures in the high dimensional regime where the dimensionality of the covariate space is allowed to grow nearly exponentially in the sample size. Our method also adapts to the unknown smoothness and sparsity levels, and can be implemented by making minimal modifications to existing Bayesian additive regression tree algorithms. 

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5. Potential theory and quasisymmetric maps between compact Ahlfors regular metric measure spaces via Besov functions: preliminary

   Lehrback, Juha; Shanmugalingam, Nageswari (May 2024, Pute and Applied Functional Analysis)

   We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space.This approach is applicable even if the space does not support any Poincar´e inequalities. As an application of the Besov capacity estimates we show that if a homeomorphism between two Ahlfors regular metric mea- sure spaces preserves, under some additional assumptions, certain Besov classes, then the homeomorphism is necessarily a quasisymmetric map. 

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