Compositionality Emerges in a Narrow Depth-Connectivity Regime: Architecture Constraints and Solution Manifolds

Compositionality is believed to be the foundation for generalization, enabling models to reuse meaningful primitives in novel combinations. Yet, models trained with standard gradient-based optimization rarely, and often only weakly, exhibit compositional internal structure, and it remains unclear how or why such compositionality forms. In this work, we show that compositionality emerges in a narrow connectivity-depth sweet spot. Along the connectivity axis, compositionality only appears in some specifically sparse networks, heavily depends on which connections remain rather than on weights' sparsity alone. Along the depth axis, compositionality emerges within a narrow, target-dependent regime, peaking at specific depths, while both shallower and deeper networks fail. When either the depth or connectivity condition is violated, gradient descent silently converges to fractured solutions rather than compositional ones. To discover and exploit this emergence, we introduce (i) similarity-based pruning (SP) to recover compositional connectivity and (ii) a heuristic depth predictor to estimate where compositionality is most likely to appear. Finally, we support these empirical findings with a theoretical framework based on compositional sparsity, volume-ratio arguments, and feature-interference bounds, explaining why compositional solutions are reachable only in a narrow depth-connectivity regime.

Compositional Boundaries for Density Fusion

Distributed uncertainty-management systems often combine local probabilistic models along aggregation trees chosen by communication, privacy, or scheduling constraints. The final density should depend on the weighted sources, not on the particular order in which intermediate nodes combine them. We study this requirement as an algebraic compositionality problem for binary fusion of weighted probability densities. The central question is when a local fusion rule can be executed hierarchically while remaining order-invariant. We establish a compositional boundary for local segment-valued fusion rules. Within the class of continuous binary rules with additive output weights and weight-only coefficients, order-invariant hierarchical execution characterizes normalized weighted linear pooling; norm-induced segment balancing realizes the corresponding coefficient. Smooth endpoint-to-candidate $f$-divergence balancing has a different local geometry: its quadratic expansion induces square-root effective weights, showing why pairwise solvability alone is insufficient for schedule-independent fusion. We show that this obstruction is local to endpoint-to-candidate binary balancing, whereas global divergence barycenters retain additive-weight local limits. Finally, Gaussian mixtures show how the same issue appears in finite model classes: exact fusion is compositional, whereas stepwise compression is compositional only under a congruence condition on unnormalized component measures. These results distinguish exact schedule-independent fusion from global aggregation objectives and local approximation heuristics.