DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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Equivariant neural layers between group-action feature spaces

Definition

Let a group $G$ act on sets $X$ and $Y$ by left actions $T$ and $T^{'}$ , respectively. A layer is a function $Φ : X \to Y$ . It is $G$ -equivariant when

Φ (T_{g} x) = T_{g}^{'} Φ (x) (g \in G, x \in X) .

The input and output actions need not be the same. The layer is $G$ -invariant when $Φ (T_{g} x) = Φ (x)$ for every $g \in G$ and $x \in X$ ; equivalently, it is equivariant into $Y$ equipped with the trivial action.

Definition references

equivariant maps between acted-on sets

the induced action on feature maps

functions and declared domains/codomains

Cohen and Welling, “Group Equivariant Convolutional Networks,” §2

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

Depends on

Used by

Group cross-correlation and compatible layers are equivariant Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)