Pruning at Initialisation through the lens of Graphon Limit: Convergence, Expressivity, and Generalisation

Pruning at Initialisation methods discover sparse, trainable subnetworks before training, but their theoretical mechanisms remain elusive. Existing analyses are often limited to finite-width statistics, lacking a rigorous characterisation of the global sparsity patterns that emerge as networks grow large. In this work, we connect discrete pruning heuristics to graph limit theory via graphons, establishing the graphon limit of PaI masks . We introduce a Factorised Saliency Model that encompasses popular pruning criteria and prove that, under regularity conditions, the discrete masks generated by these algorithms converge to deterministic bipartite graphons. This limit framework establishes a novel topological taxonomy for sparse networks: while unstructured methods (e.g., Random, Magnitude) converge to homogeneous graphons representing uniform connectivity, data-driven methods (e.g., SNIP, GraSP) converge asymptotically to heterogeneous graphons that encode implicit feature selection. Leveraging this continuous characterisation, we derive two consequences. First, we prove a universal approximation theorem for sparse networks on active coordinate subspaces. Second, under the Graphon-NTK lazy-training regime, we connect the limiting graphon to NTK-style generalisation bounds and introduce a path-density interpretation of how sparse topology can modulate kernel alignment. Our results transform the study of sparse neural networks from combinatorial graph problems into a rigorous framework of continuous operators, offering a new mechanism for analysing expressivity and generalisation in sparse networks.