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

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📄 Abstract

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.

Authors (5)

Benjamin Walker

Lingyi Yang

Nicola Muca Cirone

Cristopher Salvi

Terry Lyons

Submitted

May 23, 2025

arXiv Category

cs.LG

Proceedings of the Thirty-Ninth Annual Conference on Neural Information Processing Systems, 2025

arXiv PDF

Key Contributions

This paper introduces Structured Linear CDEs (SLiCEs), a unifying framework for sequence models that uses structured, input-dependent state-transition matrices. SLiCEs achieve maximal expressivity comparable to dense matrices while being computationally cheaper. The framework encompasses existing architectures and introduces novel variants, proving that block-diagonal, sparse, or Walsh-Hadamard matrices match dense matrix expressivity, unlike diagonal matrices used in models like Mamba.

Business Value

Offers more efficient and expressive models for sequential data, leading to better performance in tasks like time series forecasting, natural language understanding, and control systems. This can improve accuracy and reduce computational costs.

Paper Metadata

Innovation Type

Algorithmic Framework

Deployment Feasibility

Moderate. The framework is general and encompasses existing architectures, suggesting good adaptability. Performance gains on specific benchmarks indicate practical potential, but implementation details and computational requirements need careful evaluation.

Limitations Addressed

Computational cost of dense state-transition matrices in sequence models,Limited expressivity of diagonal state-transition matrices,Challenges in length generalization for parallel-in-time models

Performance Gains

solve A5 benchmark with a single layer,best-in-class length generalization on regular language tasks,match performance of log neural controlled differential equations

Technical Tags

Controlled Differential Equations (CDEs)Structured Linear CDEs (SLiCEs)sequence modelsstate-transition matricesmaximal expressivityparallel-in-timelength generalizationregular language tasksblock-diagonalWalsh-Hadamard transform

Research Topics

Sequence ModelingDeep Learning ArchitecturesDifferential Equations in MLTime Series AnalysisGenerative Models

Methods & Architectures

Structured Linear CDEs (SLiCEs)Block-diagonal matricesSparse matricesWalsh-Hadamard transform matrices Structured Linear CDEs (SLiCEs)Recurrent Neural Networks (RNNs)Neural Controlled Differential Equations (nCDEs)

Applications & Tasks

Time Series Forecasting Natural Language Processing Robotics Signal Processing Modeling sequential dataAchieving maximal expressivity with efficient computationImproving length generalization Sequence ModelingTime Series AnalysisLanguage Modeling

Datasets & Benchmarks

Benchmarks

A5 state-tracking benchmark • regular language tasks • six multi-step forecasting tasks

performance on A5 benchmarklength generalization performanceperformance on multi-step forecasting

Related Fields

Deep LearningDifferential EquationsTime Series AnalysisRecurrent Neural Networks

Keywords

Structured Linear CDEsSLiCEssequence modelsstate-transition matricesmaximal expressivityparallel-in-timelength generalizationcontrolled differential equationsRNNsblock-diagonalWalsh-Hadamardtime series

Academic Context

#Sequence Modeling#Deep Learning Architectures#Differential Equations in ML#Time Series Analysis#Generative Models

Commercial Potential

Potential Products

Advanced time series forecasting toolsNext-generation NLP modelsEfficient control systems

Target Industries

FinanceTechnologyRoboticsManufacturingHealthcare

Use Case Examples

Predicting stock prices with high accuracy and long-term generalizationDeveloping more capable language models for translation and summarizationControlling robotic arms with precise and adaptive movements

Competitive Edge

Provides a unifying framework that matches the expressivity of dense matrices with computational efficiency, outperforming existing parallel-in-time models on length generalization tasks.

Market Opportunity

Large and growing market for sequence modeling solutions.

Revenue Models

Licensing of the SLiCEs frameworkdevelopment of specialized sequence models.

Resource Requirements

Compute Needs

Moderate to High (depending on model size and sequence length)

Data Requirements

Sequential data, time series data

Deployment Constraints

Requires efficient implementation of CDE solvers and structured matrix operations.

Scalability

The structured matrices aim to improve computational efficiency and scalability compared to dense alternatives.

Production Readiness

Maturity Level

Research

Time to Market

2-3 years

Patent Potential

Moderate

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