First-layer and group-to-group cross-correlation

Let $G$ be a discrete group acting on the lattice ${\mathbb{Z}}^2$ on the left. Let $C_{\mathrm{i}\mathrm{n}},C_{\mathrm{o}\mathrm{u}\mathrm{t}}$ be positive integers, and regard channel vectors as row vectors. For a feature map $f:G\to {\mathbb{R}}^{C_{\mathrm{i}\mathrm{n}}}$ and a filter $\mathrm{\Psi }:G\to {\mathbb{R}}^{C_{\mathrm{i}\mathrm{n}}\times C_{\mathrm{o}\mathrm{u}\mathrm{t}}}$, the group-to-group cross-correlation is the map $C_{\mathrm{\Psi }}f:G\to {\mathbb{R}}^{C_{\mathrm{o}\mathrm{u}\mathrm{t}}}$ defined by

$$(C_{\mathrm{\Psi }}f)(g)=\sum _{h\in G}f(h)\mathrm{\Psi }(g^{-1}h)\mathrm{.}$$

For an image feature map $f:{\mathbb{Z}}^2\to {\mathbb{R}}^{C_{\mathrm{i}\mathrm{n}}}$ and a lattice filter $\mathrm{\Psi }:{\mathbb{Z}}^2\to {\mathbb{R}}^{C_{\mathrm{i}\mathrm{n}}\times C_{\mathrm{o}\mathrm{u}\mathrm{t}}}$, the first-layer image-to-group cross-correlation is the map $E_{\mathrm{\Psi }}f:G\to {\mathbb{R}}^{C_{\mathrm{o}\mathrm{u}\mathrm{t}}}$ defined by

$$(E_{\mathrm{\Psi }}f)(g)=\sum _{x\in {\mathbb{Z}}^2}f(x)\mathrm{\Psi }(g^{-1}\cdot x)\mathrm{.}$$

Each displayed sum is required to have finite support or to converge absolutely in every output coordinate. The first operation takes group-indexed features and filters; the second takes lattice-indexed image features and filters and returns group-indexed features. These are cross-correlations, with the filter argument displayed as written; they need not be commutative, and no finiteness of $G$ is assumed.

groups and their multiplication and inverses

left actions on feature maps

discrete cross-correlation and channel mixing

functions and declared domains/codomains

Cohen and Welling, “Group Equivariant Convolutional Networks,” §6.1, Eqs. (10)–(12)

Goodfellow, Bengio, and Courville, Deep Learning, Chapter 9 §§9.1–9.2 and §9.5