Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917038163755008 |
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| author | Walker, Benjamin Yang, Lingyi Cirone, Nicola Muca Salvi, Cristopher Lyons, Terry |
| author_facet | Walker, Benjamin Yang, Lingyi Cirone, Nicola Muca Salvi, Cristopher Lyons, Terry |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_17761 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models Walker, Benjamin Yang, Lingyi Cirone, Nicola Muca Salvi, Cristopher Lyons, Terry Machine Learning 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. |
| title | Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2505.17761 |