A Hypertoroidal Covering for Perfect Color Equivariance

When the color distribution of input images changes at inference, the performance of conventional neural network architectures drops considerably. A few researchers have begun to incorporate prior knowledge of color geometry in neural network design. These color equivariant architectures have modeled hue variation with 2D rotations, and saturation and luminance transformations as 1D translations. While this approach improves neural network robustness to color variations in a number of contexts, we find that approximating saturation and luminance (interval valued quantities) as 1D translations introduces appreciable artifacts. In this paper, we introduce a color equivariant architecture that is truly equivariant. Instead of approximating the interval with the real line, we lift values on the interval to values on the circle (a double-cover) and build equivariant representations there. Our approach resolves the approximation artifacts of previous methods, improves interpretability and generalizability, and achieves better predictive performance than conventional and equivariant baselines on tasks such as fine-grained classification and medical imaging tasks. Going beyond the context of color, we show that our proposed lifting can also extend to geometric transformations such as scale.

💡 Research Summary

The paper addresses a fundamental weakness of conventional convolutional neural networks (CNNs): their performance degrades sharply when the color distribution of input images changes between training and inference. Recent “color‑equivariant” architectures have attempted to mitigate this by explicitly modeling the geometry of the Hue‑Saturation‑Luminance (HSL) color space. They treat hue as a cyclic group (2‑D rotations) and model saturation and luminance as 1‑D translations on the real line. While this yields robustness to hue shifts, approximating the bounded interval‑valued saturation and luminance with an unbounded translation group introduces clipping artifacts and only approximate equivariance.

The authors propose a fundamentally different approach: they lift the bounded intervals of saturation and luminance onto the unit circle using a double‑covering map, thereby endowing these channels with a true cyclic group structure. Concretely, for a saturation interval I =