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p4 and p4m orientation-indexed feature maps

Example

Let $ρ = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ be the quarter-turn, $σ = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ a reflection, $e_{1} = (1, 0)$ , and $e_{2} = (0, 1)$ ; define $δ_{a} (x) = 1$ when $x = a$ and $0$ otherwise. The ordinary planar cross-correlation $(f ⋆ ψ) (x) = \sum_{y \in Z^{2}} f (y) ψ (y - x)$ with $f = δ_{0}$ and $ψ = δ_{e_{1}}$ is not rotation-equivariant with a fixed filter: $L_{ρ} f = f$ , but $f ⋆ ψ = δ_{- e_{1}}$ and $L_{ρ} (f ⋆ ψ) = δ_{- e_{2}} \neq δ_{- e_{1}} = (L_{ρ} f) ⋆ ψ$ . In a p4 group layer, the feature map is indexed by pairs $(t, ρ^{j})$ , giving four orientation indices at each translation $t \in Z^{2}$ ; in a p4m group layer it is indexed by $(t, ρ^{j} σ^{ϵ})$ for $ϵ \in {0, 1}$ , giving eight. The full p4 and p4m groups remain infinite through their translations, and their group-correlation layers are equivariant for the full left action.

Example references

the p4 and p4m groups and their finite point groups

the action on spatial and group-indexed maps

planar cross-correlation convention

group-correlation layer

group-correlation equivariance

Cohen and Welling, “Group Equivariant Convolutional Networks,” §§4.2–4.4, 5–6.1 and Appendix A

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

Facts & Assumptions

Given: The p4/p4m groups, feature-map actions, planar correlation, and group-correlation theorem described in the Example. Verify the orientation counts, group-layer equivariance, and fixed-filter rotation failure.

[F1]

The p4 and p4m square-lattice symmetry groups: The point groups are $C_{4} = {ρ^{j}}$ and $D_{4} = {ρ^{j}, ρ^{j} σ}$ ; the full groups contain every translation and are infinite.

[F2]

The left action on feature maps induced by an action on sites: The action on a feature map is $(L_{u} f) (x) = f (u^{- 1} \cdot x)$ .

[F3]

Discrete convolutional layer with local receptive fields, channel mixing, stride, and padding: On the full lattice with unit stride, a finite receptive-field layer computes a discrete cross-correlation without kernel reversal.

[F4]

First-layer and group-to-group cross-correlation: Group-to-group cross-correlation uses $(C_{Ψ} f) (g) = \sum_{h \in G} f (h) Ψ (g^{- 1} h)$ .

[F5]

Group cross-correlation and compatible layers are equivariant: Group-to-group cross-correlation commutes with the left action whenever its sum is finite or absolutely convergent.

Verification

1.1givenF1algebra

At a fixed translation $t$ , the p4 index has the four values $ρ^{j}$ and p4m has the eight distinct values $ρ^{j} σ^{ϵ}$ , by [F1]. Distinct translations give infinitely many full group elements in each case.

1.2givenF2F4F5

For either full group $G = P_{C_{4}}$ or $P_{D_{4}}$ , apply [F5] to the group-correlation operator in [F4]. Its input and output actions are the left actions in [F2], so every such group layer is equivariant without treating $G$ as finite.

2.1givenF2F3algebra∎

Since $ρ (0) = 0$ , $L_{ρ} δ_{0} = δ_{0}$ . In the displayed planar correlation, only $y = 0$ contributes, and $δ_{e_{1}} (e_{1}) = 1$ , so $δ_{0} ⋆ δ_{e_{1}} = δ_{- e_{1}}$ . Therefore $L_{ρ} (δ_{0} ⋆ δ_{e_{1}}) = δ_{ρ (- e_{1})} = δ_{- e_{2}}$ , whereas $(L_{ρ} δ_{0}) ⋆ δ_{e_{1}} = δ_{- e_{1}}$ . As $e_{1} \neq e_{2}$ , these outputs differ.

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Sources

Taco S. Cohen and Max Welling, Group Equivariant Convolutional Networks (standard reference, not scraped)
Ian Goodfellow, Yoshua Bengio, and Aaron Courville, Deep Learning (standard reference, not scraped)