MATH.13:4.4 - Preserve the information needed for use

When symmetry reduces a calculation, state what the reduced answer means in the original problem.

An invariant output can often be calculated from a representative of a symmetry class. An equivariant output may need the transformation used to choose that representative so the answer can be returned to the original coordinates. If several transformations give the same representative, their different output actions must agree on the returned answer, or the result remains ambiguous.

For example, rotating an image to a standard orientation can help classify the object. Reporting a location in the original image additionally requires the corresponding inverse coordinate transformation. If the task distinguishes orientations, the normalization must retain that information.

Use C.29.1 when establishing this representation-and-return relation is itself the difficulty. Use C.29.2 to construct the actual computation; the existence of a symmetric formulation does not supply an efficient algorithm.