Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Walker, Benjamin, Yang, Lingyi, Cirone, Nicola Muca, Salvi, Cristopher, Lyons, Terry
Institutions: University of Oxford, Imperial College London
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G1309768829, G1600321425, G1633672591, G3481878861, G3855435662, G444630160, The Alan Turing Institute, EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation, G7723329525, Unparameterised multi-modal data, high order signatures, and the mathematics of data science
OpenAlex ID: W6979244958
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