Natural Filter Equivariants (NFEs)

• Natural Filter Equivariants are operators that preserve symmetry by commuting with both map and filter actions across mathematical structures.
• They enable robust observer designs on manifolds and improve extrapolation in list-based models through techniques like amalgamation.
• NFEs offer algebraic and geometric frameworks that enhance system stability and error consistency in nonlinear control and deep learning.

Natural Filter Equivariants (NFEs) are a unifying conceptual framework that formalizes symmetry-preserving operators across multiple mathematical and applied domains, including nonlinear observer design, list-function extrapolation, and equivariant convolutional networks. In all settings, “natural filter equivariance” characterizes operators that commute with canonical “filter” perturbation actions—removal, restriction, or transformation—under specified group or predicate symmetries. This concept plays a crucial role in system robustness, generalization, and structure-preserving computation.

1. Core Definitions and Formal Frameworks

NFEs are rigorously defined relative to the symmetries of an underlying structure—be it a manifold, a list, or a data grid—by demanding equivariance with respect to both group actions and filter-like operations.

• On lists (Lewis et al., 11 Jul 2025), the type $[a]$ denotes all finite lists over a set $a$, equipped with:
  • Map operations: $\map(\psi):[a]\to[b]$, $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$, indexing action by $\psi: a \to b$.
  • Filter operations: $\filter(\phi):[a]\to[a]$, $\filter(\phi)[x_1, \ldots, x_n] = [x_i \mid \phi(x_i) = \mathit{true}]$, where $\phi: a \to \text{Bool}$.

A function $f: [a] \to [a]$ is filter-equivariant if $f \circ \filter(\phi) = \filter(\phi) \circ f$ for all predicates $a$0. It is a natural filter-equivariant (NFE) if, in addition, $a$1 for all $a$2.

• For nonlinear systems (Mahony et al., 2021, Mahony et al., 2020), consider a control-affine system $a$3 with Lie group symmetry $a$4:
  • System: $a$5, $a$6, $a$7, $a$8, $a$9.
  • $\map(\psi):[a]\to[b]$0 acts transitively on $\map(\psi):[a]\to[b]$1 ($\map(\psi):[a]\to[b]$2), and on inputs ($\map(\psi):[a]\to[b]$3). $\map(\psi):[a]\to[b]$4 is equivariant: $\map(\psi):[a]\to[b]$5.

An NFE is then an observer whose state evolves on $\map(\psi):[a]\to[b]$6, with dynamics, innovation, and update all commuting with group actions—in effect, a “natural” lift of filtering to a symmetry-respecting context.

2. Structural Properties and Algebraic Relationships

NFEs exhibit distinct algebraic closures and intersection properties:

• The classes of map-equivariant ($\map(\psi):[a]\to[b]$7), filter-equivariant ($\map(\psi):[a]\to[b]$8), and natural filter-equivariant ($\map(\psi):[a]\to[b]$9) functions satisfy:

$\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$0

Concrete examples include: - $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$1 (identity) and $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$2 are in $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$3. - Sorting (with fixed ordering) is in $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$4. - The “triangle” function is in $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$5. - Expanding (inflate) and reverse are canonical NFEs.

• In nonlinear systems observer theory, the NFE formalism corresponds to a class of symmetry-respecting observers (Equivariant Filters or EqF) with:
  • Error, innovation, and update performed equivariantly in the Lie group.
  • The error variable defined globally and intrinsically on the state manifold, avoiding coordinate patches.

3. Geometric and Categorical Interpretation

NFEs arise naturally in the language of category theory and geometric structures.

For list functions (Lewis et al., 11 Jul 2025):

• The assignment $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$6 is a list functor $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$7, with map- and filter-induced natural transformations.
• NFEs are characterized as doubly-natural endomorphisms: they commute with both map and filter-induced transformations, i.e., are “symmetric” with respect to all canonical list symmetries.
• There is a simplicial and multisimplicial geometric account: $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$8-NFEs correspond to cones over a semi-simplicial diagram of symmetric group actions, yielding a classification by compatible families of list permutations or multisets.

For nonlinear geometric systems (Mahony et al., 2021):

• The observer state is lifted to the symmetry group, and equivariant error is defined as $\map(\psi)[x_1, \ldots, x_n] = [\psi(x_1), \ldots, \psi(x_n)]$9, a globally well-defined quantity on $\psi: a \to b$0.
• This construction embeds the system in a natural principal bundle and leverages the geometry of group actions.

4. Methodological Realizations

NFEs have explicit constructive realizations in multiple technical domains:

Observer/Filter Design on Manifolds and Lie Groups

• An NFE observes trajectories using a lifted internal model:
  • Equivariant lift $\psi: a \to b$1 satisfying pre-image and equivariance conditions.
  • Filter equations are implemented in the group:

  $\psi: a \to b$2 - Covariance dynamics are Riccati equations with explicit curvature corrections accounting for manifold geometry.

Filter-Equivariant Functions and List Amalgamation

• Any NFE on lists can be reconstructed from its action on all one-element-filtered sublists via the amalgamation algorithm (Lewis et al., 11 Jul 2025). This operation inductively builds the output from consistent filtered restrictions: $\psi: a \to b$9

• The class of all FEs and NFEs forms a monoid under concatenation and composition.

Equivariant Filter Parameterization for Convolution

• NFEs are instantiated as “atomic basis” expansions for rotation-equivariant filters (Xie et al., 2021):
  • Any discrete, unrotated filter is exactly represented as a sum of atomic steerable bases $\psi: a \to b$3, $\psi: a \to b$4, maintaining equivariance in the continuous domain.
  • Fence-effect (aliasing) is mitigated by folding high frequencies down to within the Nyquist limit on the discrete grid.
  • Discretization yields $\psi: a \to b$5 equivariance error, which vanishes for fine meshes.

5. Robustness, Performance, and Theoretical Guarantees

NFEs confer multiple structural benefits across domains:

• Global, intrinsic error coordinates: In nonlinear filtering, the error is defined on the manifold globally, removing the need for local charts and preventing EKF singularities (Mahony et al., 2021).
• Constant linearization point: Linearization occurs solely at a fixed origin, significantly reducing linearization error compared to conventional EKF schemes.
• Curvature correction: The observer Riccati dynamics include a curvature term that tracks parallel-transport of the covariance, improving consistency, especially during transients.
• Local (and in special cases global) convergence guarantees: Under mild observability hypotheses, NFEs yield observers with provable stability (Mahony et al., 2021, Mahony et al., 2020).
• Generalization via amalgamation: For filter-equivariant sequence-to-sequence models, NFEs allow exact reconstruction on larger inputs once the function is known for small bags, implying strong extrapolation generalization (Lewis et al., 11 Jul 2025).

Experimental evidence demonstrates that, in applications such as image super-resolution, the Fourier-series NFE basis achieves state-of-the-art results, maintaining equivariance and fidelity even without data augmentation (Xie et al., 2021).

6. Connections, Extensions, and Open Problems

NFEs serve as a touchstone for symmetry-preserving design in broad contexts:

• Generalization to other combinatorial structures: NFEs defined for lists potentially extend to tree- or graph-structured data, suggesting directions for "tree-filter" or graph-filter equivariant networks (Lewis et al., 11 Jul 2025).
• Higher-order filter symmetries: The existence of “$\psi: a \to b$6-filter” operators imposing reconstructibility from larger substructures remains an open problem.
• Machine learning architectures: Implementing explicit filter-equivariant constraints as inductive biases in sequence or graph neural networks is a promising but unsolved area.
• Comparative filter-parametrizations: The “natural” filter basis, as characterized by NFE theory, outperforms limited or aliasing-prone alternatives (e.g., harmonic/PDO expansions, standard DFT) for low-level image processing tasks (Xie et al., 2021).

7. Illustrative Examples

Domain	Canonical NFE Example	Reference
Nonlinear observer	EqF on $\psi: a \to b$7: Direction estimation	(Mahony et al., 2021)
List functions	Reverse, Inflate$\psi: a \to b$8	(Lewis et al., 11 Jul 2025)
CNN filter design	Fourier-series atomic basis for F-Conv	(Xie et al., 2021)

In summary, Natural Filter Equivariants rigorously formalize the notion of symmetry-respecting transformation under both filter-like and map-like actions. They yield robust, generalizing, and expressively complete operators in observer design, algebraic function classes, and deep learning architectures, and their study illuminates the deep connections between categorical algebra, geometry, and learning theory.