Learning Compositional Latent Structure with Vector Networks

The ability of deep neural networks to learn hierarchical features is widely regarded as a key mechanism underlying their success in high-dimensional learning. Existing theory partially supports this view by establishing approximation rates based on parameter counts and sample complexity guarantees for compositional models without incurring the curse of dimensionality (CoD). To study overparameterized regimes, where the number of parameters exceeds the sample size, we develop a framework that measures complexity via the parameter norm. Within this approach, we establish approximation rates and excess risk bounds for learning sparse compositional functions whose compositional structure is represented by directed acyclic graphs (DAGs), using Frobenius norm-constrained deep neural networks. Our results have broad applicability since every function that is efficiently Turing computable admits sparse compositional representations. In particular, we cover a range of representative models, including multi-index models, binary tree structures, and general compositional architectures. The rates we derive show that deep networks can exploit the compositional structure of the target functions, effectively avoiding the CoD through hierarchical representations.

Identifying the structural priors that enable Deep Neural Networks (DNNs) to overcome the curse of dimensionality is a fundamental challenge in machine learning theory. Existing literature suggests that effective high-dimensional learning is driven by compositional sparsity, where target functions decompose into constituents supported on low-dimensional variable subsets. To investigate this hypothesis, we combine Information Filtering Networks (IFNs), which extract sparse dependency structures via constrained information maximisation, with Homological Neural Networks (HNNs), which map the inferred topology into fixed-wiring sparse neural graphs. We formalise the design principles underlying this construction and present an interpretable pipeline in which abstraction emerges through hierarchical composition. HNNs are orders of magnitude sparser than standard DNNs and require only minimal hyperparameter tuning. On synthetic tasks with known sparse hierarchies, HNNs recover the underlying compositional structure and remain stable in regimes where dense alternatives degrade as dimensionality increases. Across a broad suite of real-world datasets, HNNs consistently match or outperform dense baselines while using far fewer parameters, exhibiting lower variance and showing reduced sensitivity to hyperparameters.

In the mid-twentieth century, mathematician and polymath John von Neumann created a computational system on an array of cells as a simple model of the human brain, where each cell had one of a finite set of roles or states that he predicted would be modelled by a diffusion process. In this work, we show that such a system, when developed in a modern deep learning setting, enables the construction of an artificial neuron having specialized roles that can be learnt. We refer to this neuron as the Von Neumann neuron, and the resulting neural network from such neurons result in a self-engineered design whose architecture is only dependent on the structure and locations of its inputs and outputs on this cellular array. The mathematical framework for these Von Neumann Networks (VNNs) is also constructed and shows that they are based on the extension of neural operators and the learning of Green's functions with convolutions on a cellular topology having a diffusion signature. We also prove that these VNNs are part of a more general computational system called Cellular Machines that are computationally universal. Initial experiments show that VNN based multi-layered perceptrons outperform their equivalent deep learning variant on basic tasks, while being more parameter efficient and are capable of learning new types of tasks. This includes the ability to solve for and construct an extension of the Von Neumann (hardware) architecture common to all modern computers to cells and suggests new opportunities that could be explored.