Full characterization of the induced function class

Characterize the induced function class of random mappings produced by the stochastic graph neural network whose architecture and weights are generated from a latent anisotropic Gaussian random field on a compact, boundaryless, multiply-connected manifold.

Background

Section 10 defines an expressive stochastic capacity notion for the model class of random mappings generated by the proposed architecture and presents initial structural results. However, the authors state that a full characterization of the induced function class remains unresolved.

The open problem asks for a precise description of the model’s induced function space, including its structure and properties, to formalize the representational capabilities of the geometry-driven stochastic architecture.

References

Several deeper questions, such as the full characterization of the induced function class, identifiability of the generative hyperparameters, and convergence properties of the supervised learning estimator, remain open. These are mathematically subtle and require further development.

— Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifolds  (2512.10407 - Soize, 11 Dec 2025) in Section 10, first paragraph