Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models
Benjamin Walker, Lingyi Yang, Nicola Muca Cirone, Cristopher Salvi, Terry Lyons
Structured Linear Controlled Differential Equations (SLiCEs) provide a unifying framework for sequence models with structured, input-dependent state-transition matrices that retain the maximal expressivity of dense matrices whilst being cheaper to compute. The framework encompasses existing architectures, such as input-dependent block-diagonal linear recurrent neural networks and DeltaNet's diagonal-plus-low-rank structure, as well as two novel variants based on sparsity and the Walsh-Hadamard transform. We prove that, unlike the diagonal state-transition matrices of S4 and Mamba, SLiCEs employing block-diagonal, sparse, or Walsh-Hadamard matrices match the maximal expressivity of dense matrices. Empirically, SLiCEs solve the $A_5$ state-tracking benchmark with a single layer, achieve best-in-class length generalisation on regular language tasks among parallel models, and match the state-of-the-art performance of log neural controlled differential equations on six multivariate time-series classification datasets while cutting the average time per training step by a factor of twenty.