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Convolution and finite-group equivariance — Examples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Full-lattice translation equivariance and a zero-padding boundary failure

Example

Let $f : Z^{2} \to R$ have finite support, let $K \subseteq Z^{2}$ be finite, and choose real weights $w_{δ}$ for $δ \in K$ . The full-lattice layer $C_{w} f (p) = \sum_{δ \in K} f (p + δ) w_{δ}$ commutes with every translation $L_{t} f (p) = f (p - t)$ . A finite-grid boundary can fail: take $D = {0, e_{1}}$ with $e_{1} = (1, 0)$ , let $τ$ swap the two sites and let $T f = f \circ τ^{- 1}$ , extend arrays by zero outside $D$ , and define $C_{0} f (x) = \overline{f} (x - e_{1})$ for $x \in D$ . For $f (0) = 1$ and $f (e_{1}) = 0$ , one obtains $T (C_{0} f) = (1, 0)$ but $C_{0} (T f) = (0, 0)$ , so this zero-padded layer is not equivariant for the swap action on the two-site grid.

Example references

discrete convolutional layers and boundary extensions

the action induced on feature maps

full-lattice translation equivariance

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

Cohen and Welling, “Group Equivariant Convolutional Networks,” §§5–6.1

Facts & Assumptions

Given: The full-lattice layer and two-site zero-padded layer in the Example. Verify the translation identity and the finite boundary failure.

[F1]

Discrete convolutional layer with local receptive fields, channel mixing, stride, and padding: A discrete layer has finite receptive field and computes a weighted sum of shifted input values, using its chosen extension outside the input domain.

[F2]

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

[F3]

Group cross-correlation and compatible layers are equivariant: The full-lattice unit-stride cross-correlation is equivariant under the translation group when its sum is finite or absolutely convergent.

Verification

1.1givenF1F3

The formula $C_{w} f (p) = \sum_{δ \in K} f (p + δ) w_{δ}$ is the first-layer translation correlation with finite filter support, so [F3] gives $C_{w} (L_{t} f) = L_{t} (C_{w} f)$ for every $t \in Z^{2}$ .

2.1givenF1F2algebra∎

Since $τ$ swaps $0$ and $e_{1}$ , $T f = (0, 1)$ . Zero extension gives $C_{0} f (0) = \overline{f} (- e_{1}) = 0$ and $C_{0} f (e_{1}) = \overline{f} (0) = 1$ , so $C_{0} f = (0, 1)$ and $T (C_{0} f) = (1, 0)$ . In turn, $C_{0} (T f) (0) = \overline{T f} (- e_{1}) = 0$ and $C_{0} (T f) (e_{1}) = T f (0) = 0$ , so $C_{0} (T f) = (0, 0)$ . These outputs differ, proving the stated failure.

ExampleConstruction: AI-generatedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

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.

ExampleConstruction: AI-generatedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Non-strided, subgroup, and coset pooling on p4 features

Example

Let $G = P_{C_{4}}$ be the p4 group and let $H = {(0, ρ^{j}) : j = 0, 1, 2, 3}$ be its four-element rotation subgroup. For a scalar group feature map $F : G \to R$ , max pooling with $U = {(0, I), (0, ρ)}$ and no subsampling remains $G$ -equivariant. Restricting $F$ to $H$ retains only $H$ -equivariance. Left-coset pooling $Q_{H} F (g H) = \max_{h \in H} F (g h)$ has one value for each translation position: $(t, ρ^{r}) H = {(t, ρ^{r + j}) : j = 0, 1, 2, 3}$ , so the four orientation indices are pooled while the result remains equivariant on $G / H$ . In the separate finite case $H = G$ , $G / H$ has one element and the pooled value is invariant.

Example references

pooling and subgroup operations

the p4 group and its rotation subgroup

subgroup identity and closure conditions

pooling and subgroup equivariance conclusions

Cohen and Welling, “Group Equivariant Convolutional Networks,” §6.3 and Appendix A

Goodfellow, Bengio, and Courville, Deep Learning, Chapter 9 §9.3

Facts & Assumptions

Given: The p4 group, its rotation subgroup, and the pooling operators in the Example. Verify their equivariance and the quotient indexing.

[F1]

Pointwise nonlinearities, neighborhood pooling, subgroup sampling, and coset pooling: Non-strided pooling is $(P_{U} F) (g) = \max_{u \in U} F (g u)$ , subgroup sampling is restriction, and left-coset pooling is $(Q_{H} F) (g H) = \max_{h \in H} F (g h)$ .

[F2]

The p4 and p4m square-lattice symmetry groups: p4 elements are pairs $(t, ρ^{r})$ with product $(t, ρ^{r}) (s, ρ^{q}) = (t + ρ^{r} s, ρ^{r + q})$ .

[F3]

Group cross-correlation and compatible layers are equivariant: Non-strided pooling commutes with the left action, subgroup restriction preserves the subgroup action, and finite coset pooling is equivariant on the coset set.

[F4]

Subgroup: A subgroup contains the identity and is closed under the group operation and inverses.

Verification

1.1givenF1F3

The set $U$ is finite and nonempty, so the non-strided max pool is defined; [F3] gives $P_{U} (L_{a} F) = L_{a} (P_{U} F)$ for every $a \in G$ .

1.2givenF2F3F4algebra

The p4 multiplication in [F2] gives $(0, I) \in H$ , $(0, ρ^{j}) (0, ρ^{k}) = (0, ρ^{j + k}) \in H$ , and $(0, ρ^{j})^{- 1} = (0, ρ^{- j}) \in H$ , so [F4] confirms $H \leq G$ . For $a, h \in H$ , $a^{- 1} h \in H$ , and restricting $L_{a} F$ to $H$ equals applying the left $H$ -action to $F ∣_{H}$ . The theorem therefore retains exactly the subgroup equivariance claimed after restriction.

1.3F2algebra

By the p4 pair product in [F2], $(t, ρ^{r}) H = {(t, ρ^{r + j}) : j = 0, 1, 2, 3}$ , so each left coset fixes $t$ and includes all four rotations.

2.1givenF1F3step 1.3algebra∎

The coset-pooling theorem [F3] makes $Q_{H} F$ equivariant on $G / H$ . If $H = G$ is finite, there is one coset, and the induced left action on this singleton is the identity, so the pooled value is invariant.

Convolution and finite-group equivariance — Examples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

Full-lattice translation equivariance and a zero-padding boundary failure

Example

Example references

Facts & Assumptions

Verification

p4 and p4m orientation-indexed feature maps

Example

Example references

Facts & Assumptions

Verification

Non-strided, subgroup, and coset pooling on p4 features

Example

Example references

Facts & Assumptions

Verification

Sources