# Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifolds

arXiv: https://arxiv.org/abs/2512.10407 · HTML: https://mlsanity.com/paper/2512.10407/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 51 of its 110 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Builds on (most cited first, 30 of 51)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models](https://mlsanity.com/paper/1802.08773/) (2018)
- [GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders](https://mlsanity.com/paper/1802.03480/) (2018)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [GraphAF: a Flow-based Autoregressive Model for Molecular Graph Generation](https://mlsanity.com/paper/2001.09382/) (2020)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [SNAS: Stochastic Neural Architecture Search](https://mlsanity.com/paper/1812.09926/) (2018)
- [Efficient Low-rank Multimodal Fusion with Modality-Specific Factors](https://mlsanity.com/paper/1806.00064/) (2018)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Multiplicative Normalizing Flows for Variational Bayesian Neural Networks](https://mlsanity.com/paper/1703.01961/) (2017)
- [The Low-Rank Simplicity Bias in Deep Networks](https://mlsanity.com/paper/2103.10427/) (2021)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Bayesian Graph Neural Networks with Adaptive Connection Sampling](https://mlsanity.com/paper/2006.04064/) (2020)

## Abstract (arXiv)

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.