Equivariance

Origin domain

Mathematics

Also from

Physics

Aliases

Covariance Under a Group, Structure Preserving Transformation, Commuting with Symmetry

Core Idea

Equivariance is the structural pattern in which transforming the input of a map produces a correspondingly transformed output — the map commutes with a group of transformations rather than ignoring them. Formally, f(g·x) = g·f(x): the output does not stay fixed (that would be invariance) but changes in lockstep with the input, so the transformation can be applied before or after the map with the same result.

How would you explain it like I'm…

Move-Together Rule

Imagine a photocopier that prints exactly what you put on it. If you turn the original picture upside down, the copy also comes out upside down. The copy is not stuck in one position; it moves whenever the original moves, and it moves the same way. That kind of machine respects how you turned the picture.

Turning Together

Equivariance means: if you twist or move what goes into a machine, what comes out twists or moves in the same way. Picture a face-detector drawing a box around a person's face. If you slide the photo to the right, the box slides to the right too. The detector did not ignore your move (that would be invariance, where the box stays put), and it did not get confused. It moved along with your move.

When Inputs and Outputs Move in Lockstep

Equivariance is the property that a function commutes with a transformation: doing the transformation before the function gives the same result as doing it after. In symbols, f(g·x) = g·f(x), where g is some operation like a rotation or shift. This is different from invariance, where f(g·x) = f(x) — the output stays fixed under g. With equivariance, the output is not fixed; it transforms in lockstep with the input. A neural network that detects edges in an image is equivariant to translation: shift the image, and the detected edges shift the same amount. This is now a foundational design principle in geometric deep learning, where architectures are built to respect the symmetries of their data.

 

Equivariance is the structural pattern in which a map respects a symmetry by commuting with a group action rather than ignoring it. Formally, given a group G acting on an input space X and on an output space Y, a map f: X→Y is equivariant if f(g·x) = g·f(x) for every g in G and every x in X. The two group actions — one on inputs, one on outputs — are tied together by the map. Crucially, equivariance is distinct from invariance: invariance demands f(g·x) = f(x), where the output is unchanged by g; equivariance demands f(g·x) = g·f(x), where the output transforms compatibly. The notion descends from representation theory, where an equivariant linear map between G-representations is called an intertwiner (Serre, 1977). The same idea recurs across domains under different names: as covariance in physics (laws transform predictably under coordinate changes), as natural transformation in category theory, and as the organizing principle behind convolutional neural networks (translation-equivariant) and the broader program of geometric deep learning (Bronstein et al., 2021), which treats equivariance to relevant symmetries as the central architectural principle.

Broad Use

• Mathematics: An equivariant map between G-sets; representation theory, where intertwining maps commute with group action.
• Physics: Covariance of physical laws under coordinate change — the equations transform consistently so the physics is the same in every frame.
• Machine learning: Convolutional layers are translation-equivariant (shift the image, the feature map shifts identically), the principle behind geometric deep learning.
• Signal processing: A linear time-invariant filter — delay the input, the output is delayed by the same amount.
• Robotics/vision: Pose-equivariant representations where rotating an object rotates its encoding.

Clarity

Naming equivariance separates two often-confused notions: a quantity left unchanged by a transformation (invariance) versus a quantity that tracks the transformation predictably (equivariance). It lets designers state precisely whether they want a representation to discard a symmetry or to preserve and carry it forward.

Manages Complexity

Equivariance lets one guarantee behavior across an entire orbit of transformed inputs from a single analysis of one representative, collapsing infinitely many cases into one. It bounds the design space of models and laws to those whose structure respects a known symmetry, drastically shrinking what must be learned, proven, or checked.

Abstract Reasoning

Recognizing equivariance enables reasoning by symmetry reduction (solve once, transport the solution across the group), supports building inductive bias into models, and clarifies when invariance can be derived from equivariance followed by a symmetric pooling step.

Knowledge Transfer

The physicist's covariant equations and the ML engineer's translation-equivariant network are the same structure: a map that commutes with a symmetry group. The insight that pooling an equivariant feature yields an invariant one transfers from harmonic analysis to deep learning unchanged.

Relationships to Other Primes

Parents (3) — more general patterns this builds on

• Equivariance is a kind of Invariance — Equivariance is a kind of invariance: under a coordinated transformation of input and output, the map's structural relation to the group is preserved.
• Equivariance is a kind of Symmetry — Equivariance is a specialization of symmetry that requires the map to commute with the group action rather than be fixed by it.
• Equivariance presupposes Function (Mapping) — Equivariance presupposes Function (Mapping): the equivariance property is asserted of a deterministic map between sets carrying group actions.

Path to root: Equivariance → Invariance

Not to Be Confused With

• Equivariance is not invariance: invariance means the output is unchanged under the transformation; equivariance means the output transforms correspondingly. Invariance is the special case where the output's group action is trivial.
• It is not symmetry itself (a system unchanged by a group action); equivariance is a property of a map between systems relating their two group actions.
• It is not conjugate variables, which pairs complementary descriptions; equivariance constrains how a single map respects a transformation group.