Complete theoretical characterization of the stochastic neural model

Establish a complete theoretical characterization of the stochastic neural model whose neural architecture is generated by a latent anisotropic Gaussian random field on a compact, boundaryless, multiply-connected manifold.

Background

The paper introduces a stochastic neural architecture in which neuron locations, connectivity, and synaptic weights are generated by a latent Gaussian random field defined on a compact, boundaryless, multiply-connected manifold. Although foundational properties such as well-posedness and preliminary expressive variability are developed, the authors emphasize that a full theoretical account of the model is not yet available.

This open problem concerns developing a comprehensive mathematical framework for the entire stochastic neural system, beyond preliminary results—covering its formal properties, structural behavior, and theoretical foundations needed to fully understand and analyze the geometry-driven stochastic learning approach.

References

While a complete theoretical characterization remains open, the results already establish fundamental properties, such as a preliminary analysis of the expressive variability of the induced stochastic mappings, supporting the model internal coherence and expressive potential.

— 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 1 (vii)