The authors study tree ensembles (random forests and gradient boosting) from a spectral perspective—looking at the eigenvalues and eigenvectors of the implicit kernel or smoother matrix these methods induce. They derive minimax-optimal convergence rates for random forest regression (showing the rate depends on how fast the kernel's eigenvalues decay) and use the spectral viewpoint to compress ensembles. For random forests, the leading eigenfunctions of the kernel capture the most important predictive directions; for GBMs, the leading singular vectors of the smoother matrix do the same. Training small nonlinear maps on these spectral features yields "distilled" models orders of magnitude smaller than the originals with competitive performance.
Main takeaways:
- Random forests and GBMs can be viewed through their induced kernels or smoother matrices; the eigenvalue decay of these operators governs statistical convergence rates.
- Under mild conditions, random forest regression achieves minimax-optimal rates.
- Leading eigenfunctions (for RFs) or singular vectors (for GBMs) capture the ensemble's dominant predictive structure.
- Training small models on these spectral representations compresses ensembles by orders of magnitude with minimal performance loss.
- The method outperforms existing pruning and rule-extraction techniques on real datasets.