equivariance

Equivariance is a mathematical concept describing when a map respects a symmetry described by a group action. Let G be a group acting on spaces X and Y. A function f: X → Y is G-equivariant if for every g in G and every x in X, f(g·x) = g·f(x). When X and Y are representations (vector spaces with a linear G-action), a linear map T: V → W is G-equivariant if T(g·v) = g·T(v) for all g and v.

Equivariance is distinct from invariance; a map is invariant if f(g·x) = f(x). Equivariant maps generalize this idea by allowing the action to move both the input and the output. Examples include: the identity map on any G-space, which is vacuously equivariant; projection maps from product spaces with diagonal G-action; and any linear map between G-representations that commutes with the G-action. In analysis and physics, many constructions are equivariant, such as convolutions on signals which are translation-equivariant, or more generally, group-equivariant operations that commute with the action of a symmetry group. In machine learning, group-equivariant neural networks aim to preserve symmetry by design, yielding improved generalization.

Contexts and variants: equivariance appears in algebraic topology, representation theory, and category theory. In category theory, an equivariant object is one equipped with a G-action and a compatible structure map, and equivariant maps are morphisms respecting that action. The core idea is that constructions should commute with the given symmetry.