group invariant and group equivariant functions

1. group invariant

Let \(\mathcal{X}(\Omega)\) be the set of signals on a domain \(\Omega\)
Let \(G\) be a group and let \(\rho: G \rightarrow R^{n\times n}\) be a representation of that group
(see group representations and group actions (geometric machine learning))
A function \(f: \mathcal{X}(\Omega) \rightarrow \mathcal{Y}\) is \(G\) -invariant if \(f(x) = f(\rho(g)x)\) for all \(x\in \mathcal{X}(\Omega)\) and \(g\in G\)
for example, consider the function over images which merely sums all pixel values. Then, no matter how the image is rotated, the function output is the same.

A function \(f: \mathcal{X}(\Omega) \rightarrow \mathcal{Y}\) is \(G\) -equivariant if \(f(\rho(g)x) = \rho(g)f(x)\)
example: a convolution is shift-equivariant. A (cyclic) shifted input signal results in a shifted convolution