Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models

Benjamin Walker (University of Oxford) · Lingyi Yang (University of Oxford) · Nicola Muca Cirone (Imperial College London) · Cristopher Salvi (Imperial College London) · Terry Lyons (University of Oxford)

block-diagonal linear recurrent neural networksdeltanetdense matricesdiagonal state-transition matricesdiagonal-plus-low-rank structureinput-dependent state-transition matriceslength generalisationmambamaximal expressivitymultivariate time-series classification.s4dsparsitystate-tracking benchmarkstructured linear controlled differential equationswalsh-hadamard transform

This work introduces Structured Linear Controlled Differential Equations (SLiCEs), 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 S4D 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-in-time models, and match the 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.