Invariances

Invariances are properties of a system, quantity, or model that remain unchanged under a specified set of transformations. They provide a way to recognize what must stay the same despite changes in perspective, conditions, or representation. In mathematics and the sciences, identifying invariances helps to simplify problems, classify objects, and discover conserved quantities.

Common geometric invariances include translation, rotation, reflection, and scaling. In physics, these symmetries underpin conservation laws: translational invariance yields momentum conservation, temporal invariance yields energy conservation, and rotational invariance yields angular momentum. More generally, invariances are often described in terms of symmetry groups acting on the system.

In machine learning and computer vision, invariances describe robustness of predictions to transformations of the input. Convolutional networks exploit translation invariance, while data augmentation or architectural choices aim to induce invariance to rotations, flips, or lighting changes. In statistics and estimation, invariance concerns results that are unchanged under reparameterizations or transformations of the data.

Formally, if a group G acts on a space X, a quantity f: X → Y is invariant under G when f(g.x) = f(x) for all g in G and x in X. If outputs instead transform in a consistent way, the quantity is called equivariant. Gauge invariance, a central idea in field theories, refers to local transformations leaving observables unchanged.

See also: symmetry, group theory, Noether's theorem, conservation law, equivariance.