Characterization of chaos-component sequences of equivariant piecewise-linear maps

Characterize exactly which sequences of equivariant polynomial-chaos components arise from continuous equivariant piecewise-linear maps, beyond the established injective correspondence, by identifying conditions that ensure that a square-summable sequence reconstructs an equivariant piecewise-linear map.

Background

The paper decomposes square-integrable equivariant maps into Gaussian polynomial-chaos components and proves that every continuous equivariant piecewise-linear map is uniquely determined by its sequence of components. However, the converse direction is not established: a sequence of equivariant components may be square-summable without arising from any piecewise-linear map.

Resolving this problem would provide a structural characterization of the image of the chaos transform for equivariant piecewise-linear maps. Such a characterization would also clarify the limitations of matching only finitely many Hermite components with shifted ReLU activations, since finite matching leaves the higher-degree tail uncontrolled.

References

It remains open to characterise which sequences of chaos components come from equivariant piecewise-linear maps; see \cref{cor:pl-chaos}. The transform determines each such map uniquely, but equivariance and square-summability of the components do not ensure that the resulting map is piecewise linear.

— Graded Representation Theory of Equivariant Neural Networks  (2609.25776 - Shayestehfar, 22 Sep 2026) in Section 7, “Scope and future directions,” subsection “Limitations” (referring to Corollary 2.7)

At the architectural level, the Gram criterion describes the space available in each degree for fixed lifts and readouts. A characterisation of networks with prescribed width and depth, allowing these maps to be trained, remains open here.

— Graded Representation Theory of Equivariant Neural Networks  (2609.25776 - Shayestehfar, 22 Sep 2026) in Section 7, “Scope and future directions,” subsection “Limitations”