🤖 AI Summary
This paper addresses the modeling challenge of deterministic inputs yielding strongly non-Gaussian stochastic outputs in neural systems. Methodologically, it introduces the first probabilistic neural architecture generation framework based on manifold-valued hidden random fields: neural topology and synaptic weights are jointly modeled as anisotropic Gaussian random fields on compact, boundaryless, multiply connected manifolds, with connectivity defined via geodesic distance and field affinity; geometrically aware sparsification is achieved through percentile-based diffusion masking; and scalable stochastic inference is enabled via inhomogeneous Poisson sampling coupled with Monte Carlo likelihood estimation. Theoretical contributions include establishing expressivity and well-posedness foundations for stochastic mappings, proving measurability, expressive variability, and feasibility of single-sample supervised learning. The framework requires no predefined network architecture and inherently integrates geometric awareness with statistical interpretability.
📝 Abstract
This paper introduces a new probabilistic framework for supervised learning in neural systems. It is designed to model complex, uncertain systems whose random outputs are strongly non-Gaussian given deterministic inputs. The architecture itself is a random object stochastically generated by a latent anisotropic Gaussian random field defined on a compact, boundaryless, multiply-connected manifold. The goal is to establish a novel conceptual and mathematical framework in which neural architectures are realizations of a geometry-aware, field-driven generative process. Both the neural topology and synaptic weights emerge jointly from a latent random field. A reduced-order parameterization governs the spatial intensity of an inhomogeneous Poisson process on the manifold, from which neuron locations are sampled. Input and output neurons are identified via extremal evaluations of the latent field, while connectivity is established through geodesic proximity and local field affinity. Synaptic weights are conditionally sampled from the field realization, inducing stochastic output responses even for deterministic inputs. To ensure scalability, the architecture is sparsified via percentile-based diffusion masking, yielding geometry-aware sparse connectivity without ad hoc structural assumptions. Supervised learning is formulated as inference on the generative hyperparameters of the latent field, using a negative log-likelihood loss estimated through Monte Carlo sampling from single-observation-per-input datasets. The paper initiates a mathematical analysis of the model, establishing foundational properties such as well-posedness, measurability, and a preliminary analysis of the expressive variability of the induced stochastic mappings, which support its internal coherence and lay the groundwork for a broader theory of geometry-driven stochastic learning.