Nonparametric regression on complex domains has been a challenging task as most existing methods, such as ensemble models based on binary decision trees, are not designed to account for intrinsic geometries and domain boundaries. This article proposes a Bayesian additive regression spanning trees (BAST) model for nonparametric regression on manifolds, with an emphasis on complex constrained domains or irregularly shaped spaces embedded in Euclidean spaces. Our model is built upon a random spanning tree manifold partition model as each weak learner, which is capable of capturing any irregularly shaped spatially contiguous partitions while respecting intrinsic geometries and domain boundary constraints.

These appendices provide supplementary details and results of BAST. Appendix A contains additional details on Bayesian estimation and prediction. Prediction at u is then performed as stated in Section 3.2. The experiment setup is the same as in Section 4.1 Table S3 shows the performance of BAST and BART using the hyperparameters chosen by CV (referred to as BAST -cv and BART -cv, respectively). As a benchmark, the performance metrics for BAST and BART using the hyperparameters in Section 4.1 are also included (referred to as Standard errors are given in parentheses.

In this paper, we consider nonparametric regression problems with features lying on constrained domains or, more generally, compact Riemannian manifolds.

Nonparametric regression on complex domains has been a challenging task as most existing methods, such as ensemble models based on binary decision trees, are not designed to account for intrinsic geometries and domain boundaries. This article proposes a Bayesian additive regression spanning trees (BAST) model for nonparametric regression on manifolds, with an emphasis on complex constrained domains or irregularly shaped spaces embedded in Euclidean spaces. Our model is built upon a random spanning tree manifold partition model as each weak learner, which is capable of capturing any irregularly shaped spatially contiguous partitions while respecting intrinsic geometries and domain boundary constraints. Equipped with a soft prediction scheme, BAST is demonstrated to significantly outperform other competing methods in simulation experiments and in an application to the chlorophyll data in Aral Sea, due to its strong local adaptivity to different levels of smoothness.

Accurate sound localization in a reverberation environment is essential for human auditory perception. Recently, Convolutional Neural Networks (CNNs) have been utilized to model the binaural human auditory pathway. However, CNN shows barriers in capturing the global acoustic features. To address this issue, we propose a novel end-to-end Binaural Audio Spectrogram Transformer (BAST) model to predict the sound azimuth in both anechoic and reverberation environments. Two modes of implementation, i.e. BAST-SP and BAST-NSP corresponding to BAST model with shared and non-shared parameters respectively, are explored. Our model with subtraction interaural integration and hybrid loss achieves an angular distance of 1.29 degrees and a Mean Square Error of 1e-3 at all azimuths, significantly surpassing CNN based model. The exploratory analysis of the BAST's performance on the left-right hemifields and anechoic and reverberation environments shows its generalization ability as well as the feasibility of binaural Transformers in sound localization. Furthermore, the analysis of the attention maps is provided to give additional insights on the interpretation of the localization process in a natural reverberant environment.

These appendices provide supplementary details and results of BAST. Appendix A contains additional details on Bayesian estimation and prediction. Supplementary simulation details and results including hyperparameter tuning and computation time can be found in Appendix B. Finally, Appendix C provides the proof of Proposition 1. Appendix A.1 Estimation This appendix provides details on the Markov chain Monte Carlo (MCMC) algorithm discussed in Section 3.1. This probability specification works well in our experiments, but one can modify it if desired. Appendix A.2 Prediction in Two-dimensional Constrained Domains In this subsection we provide details on specifying the neighbor set N To sample the cluster membership of u, we need to determine the cluster memberships for vertices on the domain boundary, which can be done by, for instance, assigning a boundary vertex to the same cluster as its nearest vertex in S with respect to the graph distance in the CDT mesh (when the number of vertices in the CDT graph is large, we expect this to well approximate the geodesic distance).

Nonparametric regression on complex domains has been a challenging task as most existing methods, such as ensemble models based on binary decision trees, are not designed to account for intrinsic geometries and domain boundaries. This article proposes a Bayesian additive regression spanning trees (BAST) model for nonparametric regression on manifolds, with an emphasis on complex constrained domains or irregularly shaped spaces embedded in Euclidean spaces. Our model is built upon a random spanning tree manifold partition model as each weak learner, which is capable of capturing any irregularly shaped spatially contiguous partitions while respecting intrinsic geometries and domain boundary constraints.

Automatic code summarization frees software developers from the heavy burden of manual commenting and benefits software development and maintenance. Abstract Syntax Tree (AST), which depicts the source code's syntactic structure, has been incorporated to guide the generation of code summaries. However, existing AST based methods suffer from the difficulty of training and generate inadequate code summaries. In this paper, we present the Block-wise Abstract Syntax Tree Splitting method (BASTS for short), which fully utilizes the rich tree-form syntax structure in ASTs, for improving code summarization. BASTS splits the code of a method based on the blocks in the dominator tree of the Control Flow Graph, and generates a split AST for each code split. Each split AST is then modeled by a Tree-LSTM using a pre-training strategy to capture local non-linear syntax encoding. The learned syntax encoding is combined with code encoding, and fed into Transformer to generate high-quality code summaries. Comprehensive experiments on benchmarks have demonstrated that BASTS significantly outperforms state-of-the-art approaches in terms of various evaluation metrics. To facilitate reproducibility, our implementation is available at https://github.com/XMUDM/BASTS.

Bandit based methods for tree search have recently gained popularity when applied to huge trees, e.g. in the game of go [6]. Their efficient exploration of the tree enables to re- turn rapidly a good value, and improve preci- sion if more time is provided. The UCT algo- rithm [8], a tree search method based on Up- per Confidence Bounds (UCB) [2], is believed to adapt locally to the effective smoothness of the tree. However, we show that UCT is "over-optimistic" in some sense, leading to a worst-case regret that may be very poor. We propose alternative bandit algorithms for tree search. First, a modification of UCT us- ing a confidence sequence that scales expo- nentially in the horizon depth is analyzed. We then consider Flat-UCB performed on the leaves and provide a finite regret bound with high probability. Then, we introduce and analyze a Bandit Algorithm for Smooth Trees (BAST) which takes into account ac- tual smoothness of the rewards for perform- ing efficient "cuts" of sub-optimal branches with high confidence. Finally, we present an incremental tree expansion which applies when the full tree is too big (possibly in- finite) to be entirely represented and show that with high probability, only the optimal branches are indefinitely developed. We illus- trate these methods on a global optimization problem of a continuous function, given noisy values.