Group Equivariant Non-Expansive Operators (GENEOs)

• GENEOs are symmetry-respecting, non-expansive operators that map structured data between invariant spaces while preventing signal amplification.
• They bridge topological data analysis, geometric machine learning, and graph comparison by combining stability, compactness, and aggregation techniques.
• Practical implementations include noise reduction in signals, urban wireless reconstruction, and interpretable surrogate modeling in applications like protein pocket detection.

Group Equivariant Non-Expansive Operators (GENEOs) are symmetry-respecting, distance-non-increasing operators that act on structured data spaces and formalize the idea of an observer whose transformations preserve prescribed invariances while not amplifying perturbations. In the classical function-space setting, a GENEO is a map $F:\Phi\to\Psi$ between perception pairs $(\Phi,G)$ and $(\Psi,H)$ such that $F(\varphi g)=F(\varphi)T(g)$ for a homomorphism $T:G\to H$, together with the non-expansiveness condition $$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$. In the broader perception-space formulation, a GENEO is a GEO $(f,t)$ satisfying equivariance $f(g*x)=t(g)*f(x)$ and non-expansiveness both on data and on symmetry groups. Across the recent literature, GENEOs serve as a common language linking topological data analysis, geometric machine learning, interpretable surrogate modeling, graph comparison, signal denoising, and low-complexity equivariant architectures (Bergomi et al., 2018, Colombini et al., 3 Mar 2025).

1. Foundational definitions and conceptual scope

The original GENEO framework models data as bounded real-valued functions on a common domain. A perception pair is a pair $(\Phi,G)$, where $\Phi\subseteq \mathbb{R}^X_b$ is a space of admissible measurements and $(\Phi,G)$0 is a subgroup of $(\Phi,G)$1-preserving bijections of $(\Phi,G)$2. The ambient metric on $(\Phi,G)$3 is the sup norm,

$(\Phi,G)$4

and $(\Phi,G)$5 induces on $(\Phi,G)$6 the pseudo-metric

$(\Phi,G)$7

Within this setting, a GEO satisfies

$(\Phi,G)$8

and a GENEO is a GEO that is also non-expansive in the sup norm (Bergomi et al., 2018, Frosini et al., 2022).

A broader abstraction appears in the perception-space formalism. An (extended) perception space $(\Phi,G)$9 consists of a pseudo-metric space $(\Psi,H)$0, a group $(\Psi,H)$1, and a left group action $(\Psi,H)$2 such that

$(\Psi,H)$3

This shifts the theory beyond data represented as functions and makes the symmetry action intrinsically metric. In this language, a GEO is a pair $(\Psi,H)$4 with $(\Psi,H)$5 and $(\Psi,H)$6 satisfying

$(\Psi,H)$7

while a GENEO additionally satisfies

$(\Psi,H)$8

where

$(\Psi,H)$9

In this terminology, GENEOs are a constrained subclass of GEOs rather than a separate notion (Colombini et al., 3 Mar 2025).

Two misconceptions are explicitly excluded by the literature. First, a GEO is not necessarily a GENEO: non-expansiveness is an extra requirement. Second, GENEOs are not restricted to neural network layers. Orthogonal projection from subsets of $F(\varphi g)=F(\varphi)T(g)$0 to subsets of $F(\varphi g)=F(\varphi)T(g)$1, the persistence-diagram operator, and explicit max/min denoisers on one-dimensional signals are all treated as GENEOs in specific settings (Colombini et al., 3 Mar 2025, Frosini et al., 19 Jun 2026, Frosini et al., 2022).

2. Operator-space structure, compactness, and geometry

A central object in the theory is the natural pseudo-distance

$F(\varphi g)=F(\varphi)T(g)$2

which measures similarity modulo the symmetry group. GENEOs contract this distance: if $F(\varphi g)=F(\varphi)T(g)$3 is a GENEO from $F(\varphi g)=F(\varphi)T(g)$4 to $F(\varphi g)=F(\varphi)T(g)$5, then

$F(\varphi g)=F(\varphi)T(g)$6

This places GENEOs among the stable observers of the theory and explains their role in invariant comparison (Bergomi et al., 2018).

Under compactness assumptions on $F(\varphi g)=F(\varphi)T(g)$7 and $F(\varphi g)=F(\varphi)T(g)$8, the space of GENEOs for a fixed homomorphism $F(\varphi g)=F(\varphi)T(g)$9 is compact with respect to the operator sup metric

$T:G\to H$0

If $T:G\to H$1 is convex, the same space is convex. These properties are repeatedly used as structural justification for finite approximation, operator sampling, and low-complexity model design (Bergomi et al., 2018, Bocchi et al., 12 Mar 2025).

The compactification theory extends these results beyond already compact perception pairs. If a perception pair has a totally bounded data space rich enough to endow the common domain with a metric structure, then it admits a compactification $T:G\to H$2. Likewise, if a space of GENEOs is collectionwise surjective and the data sets are totally bounded and metrize the domains, then the operator space admits a compactification compatible with the compactified source and target perception pairs. The covering condition

$T:G\to H$3

is the key hypothesis ensuring non-expansiveness of the acting homomorphism and hence extension of equivariance to the closures (Ahmad, 2022).

A further refinement replaces worst-case operator comparison by a probability-weighted $T:G\to H$4-type geometry. In the finite-dimensional probabilistic setting, the ambient space $T:G\to H$5 becomes a Hilbert manifold with inner product

$T:G\to H$6

and the full space of GENEOs is compact with respect to the induced $T:G\to H$7 norm. Any finite-dimensional $T:G\to H$8-submanifold of this Hilbert space whose points are GENEOs inherits a Riemannian structure, making gradient-based optimization over parameterized GENEO families available in principle (Cascarano et al., 2021).

3. Construction principles and representation theorems

Several papers develop explicit mechanisms for building new GENEOs from known ones. One general device is pointwise aggregation by a $T:G\to H$9-Lipschitz scalar map $$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$0. If $$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$1 are GENEOs and

$$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$2

takes values in $$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$3, then $$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$4 is again a GENEO. From this, the literature derives power-mean constructions

$$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$5

for $$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$6, and series constructions

$$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$7

under bounded-partial-sum and $$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$8 assumptions (Quercioli, 2020).

In the finite linear setting, a more algebraic picture emerges. If a finite group $$\|F(\varphi_1)-F(\varphi_2)\|_\infty\le \|\varphi_1-\varphi_2\|_\infty$$9 acts transitively on $(f,t)$0, then every linear $(f,t)$1-equivariant operator on $(f,t)$2 can be represented by a permutant measure: $(f,t)$3 where $(f,t)$4 is constant on conjugacy orbits under $(f,t)$5. The operator is a linear GENEO precisely when

$(f,t)$6

This converts abstract linear equivariance and non-expansiveness into a finite weighted sum over permutations with an $(f,t)$7-type constraint on coefficients (Bocchi et al., 2020).

The heterogeneous version replaces ordinary permutant measures by generalized $(f,t)$8-permutant measures on $(f,t)$9, where $f(g*x)=t(g)*f(x)$0 mediates between different source and target symmetry groups. Under the assumption that $f(g*x)=t(g)*f(x)$1 acts transitively on the finite output domain $f(g*x)=t(g)*f(x)$2, a linear map $f(g*x)=t(g)*f(x)$3 is a linear GENEO if and only if

$f(g*x)=t(g)*f(x)$4

for a generalized $f(g*x)=t(g)*f(x)$5-permutant measure $f(g*x)=t(g)*f(x)$6 satisfying

$f(g*x)=t(g)*f(x)$7

The space of such linear GENEOs is a compact convex polytope generated by orbit-based building blocks (Conti et al., 7 Jan 2026).

This representation viewpoint extends naturally to graphs. Generalized permutants and generalized permutant measures on spaces of maps between vertex sets or edge sets generate vertex-weighted and edge-weighted graph GENEOs, including graph-to-graph operators with different source and target graphs (Ahmad et al., 2022).

4. Persistent homology, stability, and denoising

Persistent homology is not peripheral to the theory; it is one of its native comparison mechanisms. For a function $f(g*x)=t(g)*f(x)$8, persistence diagrams and persistent Betti numbers satisfy the stability chain

$f(g*x)=t(g)*f(x)$9

and for a family $(\Phi,G)$0 of GENEOs,

$(\Phi,G)$1

defines a strongly $(\Phi,G)$2-invariant pseudo-metric. Under additional assumptions, taking the full space of GENEOs recovers the natural pseudo-distance exactly: $(\Phi,G)$3 This is one of the foundational duality results of the subject (Bergomi et al., 2018, Frosini et al., 19 Jun 2026).

The persistence-diagram operator itself can be treated as a GENEO when persistence diagrams are represented functionally and the target group is trivial. This is one reason GENEOs are described as a bridge between TDA and machine learning in the later expository literature (Frosini et al., 19 Jun 2026).

GENEOs also appear as explicit denoisers. For one-dimensional $(\Phi,G)$4-Lipschitz signals $(\Phi,G)$5 corrupted by impulsive noise, the operators

$(\Phi,G)$6

are GENEOs under Euclidean isometries of $(\Phi,G)$7. Their composition removes upward and downward impulses under thinness and separation assumptions, yielding deterministic and probabilistic control of persistence-diagram perturbations. In particular, if $(\Phi,G)$8 and $(\Phi,G)$9 are the degree-0 persistence diagrams of $\Phi\subseteq \mathbb{R}^X_b$0 and $\Phi\subseteq \mathbb{R}^X_b$1, then

$\Phi\subseteq \mathbb{R}^X_b$2

This makes GENEOs concrete noise-reduction operators rather than merely abstract symmetry constraints (Frosini et al., 2022).

5. Interpretability, surrogate modeling, and partial equivariance

A recent line of work reinterprets GENEOs as observer-controlled translations for explanation. In the perception-space formalism, an observer is a pair $\Phi\subseteq \mathbb{R}^X_b$3, where $\Phi\subseteq \mathbb{R}^X_b$4 is a category of translation GENEOs and $\Phi\subseteq \mathbb{R}^X_b$5 is a complexity assignment. Given two GEOs, a crossed pair of translation uses GENEOs to compare them through an approximately commutative diagram, and the resulting surrogate distance

$\Phi\subseteq \mathbb{R}^X_b$6

is a hemi-metric. Symmetrization yields a pseudo-metric

$\Phi\subseteq \mathbb{R}^X_b$7

Interpretability is then defined by the conjunction of approximation quality and lower complexity: $\Phi\subseteq \mathbb{R}^X_b$8 In this framework, standard fidelity appears as the identity-only special case, while equivariant surrogate learning becomes diagrammatic distance minimization (Colombini et al., 3 Mar 2025).

The partial-equivariance extension weakens the group requirement. A P-GENEO acts between perception triples $\Phi\subseteq \mathbb{R}^X_b$9 and $(\Phi,G)$00, where $(\Phi,G)$01 and $(\Phi,G)$02 are selected admissible sets of bijections rather than necessarily groups. The defining relation is

$(\Phi,G)$03

together with non-expansiveness of both $(\Phi,G)$04 and $(\Phi,G)$05. When $(\Phi,G)$06, $(\Phi,G)$07, and $(\Phi,G)$08 are groups, one recovers the ordinary GENEO notion. The theory stresses that partial equivariance is not approximate equivariance; it is equivariance with respect to a selected admissible set of transformations (Ferrari et al., 2023).

Empirical work on GENEOnet has used these ideas to support explainability and trustworthiness claims in protein pocket detection. In that setting, the model has only 17 trainable parameters, the convex combination weights $(\Phi,G)$09 act as model-specific global explanations, and statistical resampling shows that some coefficients remain stably large or near zero across repeated retraining. The same study reports that GENEOnet exhibits a significantly higher proportion of equivariance than DeepPocket, Fpocket, and P2Rank under the tested $(\Phi,G)$10 rotation around the $(\Phi,G)$11-axis, and that consecutive predictions remain highly overlapping under molecular-dynamics perturbations, especially at smaller time steps (Bocchi et al., 12 Mar 2025).

6. Applications, empirical performance, and current boundaries

GENEOs have been instantiated in several domain-specific architectures. In protein pocket detection, GENEOnet represents proteins by eight voxelized channels and applies a shallow GENEO network with channel-specific equivariant operators, convex combination, and thresholding. The resulting model has 17 trainable parameters and, on the reported benchmark, achieves $(\Phi,G)$12 and $(\Phi,G)$13, outperforming or matching several state-of-the-art baselines while using only 200 training proteins (Bocchi et al., 2022).

In sparse urban wireless reconstruction, the signal space is modeled as bounded functions on a planar domain, and the implemented GENEO is a pattern-library reconstruction operator. The method reconstructs urban SINR maps from $(\Phi,G)$14 observed pixels, under corruption levels $(\Phi,G)$15, using local pattern similarity maps

$(\Phi,G)$16

with translation equivariance and a non-expansive stability estimate. The reported evaluation uses both MSE and persistence-diagram $(\Phi,G)$17-Wasserstein distance; the paper states that GENEO is dramatically better than both 1-KNN and U-Net in $(\Phi,G)$18-Wasserstein on Munich and yields substantially lower topological error than both baselines in Paris zero-shot transfer (Amorosa et al., 25 Jul 2025).

In graph comparison, GENEOs are built from subgraph permutants. A graph is encoded as a binary function on unordered vertex pairs, the symmetry group is the full vertex-permutation group, and the operator

$(\Phi,G)$19

is a linear GENEO that measures normalized induced occurrences of a pattern graph $(\Phi,G)$20. Pairwise comparison uses

$(\Phi,G)$21

and aggregation by maxima yields a sound certificate of non-isomorphism: isomorphic graphs always have zero score, while a positive score certifies non-isomorphism. On sampled $(\Phi,G)$22-regular graph pairs, the reported GENEO-3 model reaches accuracy $(\Phi,G)$23 for $(\Phi,G)$24 at all tested $(\Phi,G)$25, with runtimes increasing from $(\Phi,G)$26 s at $(\Phi,G)$27 to $(\Phi,G)$28 s at $(\Phi,G)$29 (Bocchi et al., 2024).

The current boundaries of the theory are also explicit. Many exact representation results are linear and finite-dimensional; the compactification theory requires total boundedness, metric richness, and, for operator-space compactification, collectionwise surjectivity; the impulsive-noise denoising results are restricted to one-dimensional signals and degree-0 persistence; and the partial-equivariance formalism still assumes bijective domain transformations (Conti et al., 7 Jan 2026, Ahmad, 2022, Frosini et al., 2022, Ferrari et al., 2023). This suggests a subject that is already structurally rich, but whose strongest theorems remain concentrated in carefully controlled settings rather than in unrestricted deep-learning practice.