Abstract

Sequence modeling tasks across domains such as natural language processing, time-series forecasting, speech recognition, and control require learning complex mappings from input to output sequences. In recurrent networks, nonlinear recurrence is theoretically required to universally approximate such sequence-to-sequence functions; yet in practice, linear recurrent models have often proven surprisingly effective. This raises the question of when nonlinearity is truly required. In this study, we present a framework to systematically dissect the functional role of nonlinearity in recurrent networks -- allowing to identify both when it is computationally necessary, and what mechanisms it enables. We address the question using Almost Linear Recurrent Neural Networks (AL-RNNs), which allow the recurrence nonlinearity to be gradually attenuated and decompose network dynamics into analyzable linear regimes, making the underlying computational mechanisms explicit. We illustrate the framework across a diverse set of synthetic and real-world tasks, including classic sequence modeling benchmarks, an empirical neuroscientific stimulus-selection task, and a multi-task suite. We demonstrate how the AL-RNN's piecewise linear structure enables direct identification of computational primitives such as gating, rule-based integration, and memory-dependent transients, revealing that these operations emerge within predominantly linear dynamical backbones. Across tasks, sparse nonlinearity plays several functional roles: it improves interpretability by reducing and localizing nonlinear computations, promotes shared (rather than highly distributed) representations in multi-task settings, and reduces computational cost by limiting nonlinear operations. Moreover, sparse nonlinearity acts as a useful inductive bias: in low-data regimes, or when tasks require discrete switching between linear regimes, sparsely nonlinear models often match or exceed the performance of fully nonlinear architectures. Our findings provide a principled approach for identifying where nonlinearity is functionally necessary in sequence models, guiding the design of recurrent architectures that balance performance, efficiency, and mechanistic interpretability.