Convolutional Model Trees

TL;DR

This work introduces a forest of hyperplane-based model trees for image-based regression, providing interpretable, piecewise-linear models with a 1-to-1 correspondence between image pixels and HP/leaf coefficients. The key ideas include performing least-squares fits on image blocks, applying convolutions to HP coefficients to handle small distortions, and enforcing a tilt constraint to ensure convergence to finite trees. By weighting leaf outputs and combining many trees into a forest, the approach achieves a smooth $C^{1}$ approximation, even across block boundaries, and offers practical handling of distortions via permutation-based transforms and HP coefficient convolutions. Although presented as an ideas paper, it outlines a coherent framework that blends model trees with forest smoothing and ALN concepts, inviting future theoretical refinement and empirical validation.

Abstract

A method for creating a forest of model trees to fit samples of a function defined on images is described in several steps: down-sampling the images, determining a tree's hyperplanes, applying convolutions to the hyperplanes to handle small distortions of training images, and creating forests of model trees to increase accuracy and achieve a smooth fit. A 1-to-1 correspondence among pixels of images, coefficients of hyperplanes and coefficients of leaf functions offers the possibility of dealing with larger distortions such as arbitrary rotations or changes of perspective. A theoretical method for smoothing forest outputs to produce a continuously differentiable approximation is described. Within that framework, a training procedure is proved to converge.