Convolutional Model Trees

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.

💡 Research Summary

The paper introduces Convolutional Model Trees (CMT), a novel regression framework designed for visual data that combines the interpretability of model trees with the robustness of convolutional processing. The authors start by addressing the high dimensionality of image data, proposing a pooling step that aggregates neighboring pixel values into “group‑pixels,” thereby reducing the dimensionality from potentially millions to a manageable size.

With the reduced‑dimensional image space represented as a compact hyper‑rectangle (HR), the method recursively partitions this space using hyperplanes (HPs). For each block, a least‑squares linear fit yields coefficients α_i that quantify how the output variable changes with respect to each pixel intensity. These coefficients form a normal vector to the HP and are placed on a 2‑D grid that mirrors the pixel layout, establishing a one‑to‑one correspondence between image sensors and model parameters.

A key innovation is the application of a circularly symmetric convolution kernel directly to the HP coefficients and to the leaf‑block functions. Because the kernel is self‑adjoint (K(x,y)=K(−x,−y)), convolving either the images or the coefficients preserves inner products, allowing the model to emulate small translations, rotations, or other minor distortions without actually transforming the input images at inference time. Consequently, the computational overhead of convolution is incurred only during training; deployment remains lightweight.

To guarantee convergence, the authors replace the traditional SSE‑based split criterion with a “tilt constraint.” After selecting the most influential axis k (based on |α_k|h_k), the split hyperplane is forced to satisfy τ|α_k|h_k ≥ Σ_{i≠k}|α_i|h_i for a chosen τ∈(0,1). This ensures that each child block’s bounding box is strictly smaller along axis k, leading to progressive reduction of all dimensions. Under the assumption of an unlimited supply of random samples from a C¹ target function, the paper proves that repeated application of this constrained splitting yields leaf blocks whose linear approximations meet any prescribed RMS error ε.

The authors further extend a single CMT into a forest. By training multiple trees with different random samples or different kernel parameters, and by assigning weight functions to each leaf, the forest’s output is obtained as a weighted average of the individual tree predictions. This averaging smooths out discontinuities at block boundaries, producing a globally C¹‑continuous approximation of the target function.

The paper also discusses practical considerations such as variable elimination (removing low‑importance pixels to satisfy hardware parallelism limits) and the impact of edge effects, which are ignored in the theoretical development but would need handling in real implementations.

Overall, the contribution lies in a theoretically grounded, interpretable regression model that can handle modest image distortions through coefficient‑level convolution, guarantees convergence via a geometrically motivated tilt constraint, and achieves smooth, differentiable predictions by aggregating multiple trees. The approach promises efficient inference suitable for embedded or real‑time systems, while retaining the ability to generate new trees for transformed inputs without additional training—a form of data‑augmentation‑free robustness.