Scaleequivariant

Scaleequivariant describes a property of a function, model, or system in which a scaled input leads to a correspondingly transformed output. In mathematical terms, a feature map f defined on an input space X is scale-equivariant with respect to a scale group G if, for every scale factor a in G, there exists a representation ρ(a) such that f(a·x) = ρ(a) f(x). The group G is typically the positive real numbers under multiplication (R+), representing continuous scaling, or a discrete subset of scales.

In computer vision, scaleequivariant networks implement these ideas so that features learned at one scale match those at other scales. This is commonly achieved by constructing layer operations that are equivariant to scale transformations, using weight sharing across scales, and performing group convolutions on the scale group. Implementations may discretize scales into a finite set and use dilated or multi-scale filters to approximate continuous scale symmetry.

Benefits include improved robustness to scale variation, better generalization to unseen sizes, and reduced reliance on data augmentation. Drawbacks include increased architectural and computational complexity, discretization artifacts, and challenges in designing appropriate representations ρ for complex tasks.

Applications span image recognition, aerial or satellite imagery, biological and microscopic imaging, and any domain where objects appear at multiple sizes. The concept is related to group-equivariant neural networks and steerable filters, extending standard convolutions to operate on symmetry groups beyond translation.