# Sparse residual tree and forest

## TL;DR

This paper introduces Sparse Residual Trees and Forests, adaptive methods for multivariate data approximation that are efficient, stable, and capable of identifying regions needing data refinement, with proven convergence and promising numerical results.

## Contribution

It proposes a novel hierarchical parallel SRT algorithm combined with RBF explorations, and introduces SRF to enhance approximation accuracy and stability.

## Key findings

- Achieves $ ext{O}(N ext{log}_2N)$ time complexity for initial work
- Provides convergence results for SRTs and SRFs
- Demonstrates effectiveness through numerical experiments

## Abstract

Sparse residual tree (SRT) is an adaptive exploration method for multivariate scattered data approximation. It leads to sparse and stable approximations in areas where the data is sufficient or redundant, and points out the possible local regions where data refinement is needed. Sparse residual forest (SRF) is a combination of SRT predictors to further improve the approximation accuracy and stability according to the error characteristics of SRTs. The hierarchical parallel SRT algorithm is based on both tree decomposition and adaptive radial basis function (RBF) explorations, whereby for each child a sparse and proper RBF refinement is added to the approximation by minimizing the norm of the residual inherited from its parent. The convergence results are established for both SRTs and SRFs. The worst case time complexity of SRTs is $\mathcal{O}(N\log_2N)$ for the initial work and $\mathcal{O}(\log_2N)$ for each prediction, meanwhile, the worst case storage requirement is $\mathcal{O}(N\log_2N)$, where the $N$ data points can be arbitrary distributed. Numerical experiments are performed for several illustrative examples.