Researchers have introduced a new approach to sequence modeling called linear oscillatory state-space (LinOSS) models, designed for efficient learning on long sequences. Drawing inspiration from biological neural dynamics, LinOSS leverages forced harmonic oscillators to provide stable, time-reversible, and expressive modeling, without the need for restrictive parameterizations.
Study: Oscillatory State-Space Models. Image Credit: khunkornStudio/Shutterstock.com
In benchmark tests, it consistently outperforms leading sequence models. For example, on a task involving sequences of 50,000 elements, LinOSS achieves nearly double the performance of Mamba and 2.5 times that of linear recurrent units (LRU).
Background
State-space models (SSMs) have become promising alternatives to transformers and recurrent neural networks (RNNs) for long-sequence tasks, offering faster inference and competitive performance across domains such as language, audio, and genomics. Early SSMs used structured state matrices to model long-range dependencies via FFT-based computations. Later versions simplified these structures to diagonal matrices but were often limited by inflexible parameterizations that constrained stability and expressivity.
LinOSS addresses these challenges head-on. Inspired by dynamics in biological and physical systems, it introduces a system of forced harmonic oscillators that achieves both stability and expressiveness using only nonnegative diagonal matrices.
Core innovations include stable discretization methods, time-reversible dynamics, and parallel scan operations that boost efficiency during both training and inference. The result is a model that not only offers theoretical robustness but also surpasses top-performing alternatives like S4, Mamba, and LRU.
The LinOSS Architecture
At its foundation, LinOSS models rely on the behavior of forced harmonic oscillators. Unlike traditional state-space models that often require tightly constrained parameters, LinOSS uses a simple diagonal state matrix that guarantees stability without sacrificing modeling power. The architecture consists of stacked LinOSS blocks, each combining oscillatory state-space behavior with nonlinear transformations.
The model introduces two discretization strategies—LinOSS-IM (implicit) and LinOSS-IMEX (implicit-explicit)—which enable efficient computation through parallel scans. These methods significantly reduce training and inference times while preserving important dynamical properties like time reversibility.
Theoretical Foundations
LinOSS brings a strong theoretical backbone to the table. While many RNNs rely on nonlinearity to avoid unstable dynamics, LinOSS achieves this naturally through its oscillatory framework. By analyzing the eigenvalues of its transition matrix, the researchers demonstrated that the model maintains bounded hidden states even for very long sequences, thereby avoiding the vanishing or exploding gradients common in other models.
Stability is straightforward to enforce: as long as the diagonal weights are nonnegative, the model remains stable. This can be easily achieved using constraints like ReLU activation, allowing greater design flexibility compared to older state-space models that required more rigid configurations.
Crucially, LinOSS is provably universal, it can approximate any continuous, causal operator acting on time-varying inputs with high precision. The two main variants offer tailored benefits:
- LinOSS-IM introduces controlled dissipation, making it ideal for stable long-term forecasting.
- LinOSS-IMEX preserves energy, enabling reversible dynamics.
This versatility allows LinOSS to adapt to a wide range of tasks and system types.
Empirical Performance
Across various sequential benchmarks, LinOSS delivers standout performance. On the UEA-MTSCA benchmark for long-range classification, LinOSS-IM achieves an average accuracy of 67.8 %, outperforming Log-NCDE (64.4 %) and S5 (63.1 %). It also sets a new benchmark on the difficult EigenWorms dataset with 95 % accuracy.
For ultra-long sequences—like those found in the PPG-DaLiA dataset (~50,000 steps)—LinOSS-IM cuts prediction error by half compared to Mamba and by 2.5x compared to LRU. In long-horizon weather forecasting, both LinOSS variants outperform transformer-based and other SSM baselines.
Ablation studies highlight the model’s robustness to hyperparameter variations. ReLU parameterization slightly boosts performance. LinOSS-IMEX is especially effective in systems that conserve energy, while LinOSS-IM is better suited to dissipative systems—demonstrating the model’s adaptability.
Conclusion
LinOSS marks a notable advance in sequence modeling by combining biologically inspired design with solid theoretical underpinnings. Using harmonic oscillators, it achieves stable, efficient, and expressive learning on sequences of substantial length, without relying on restrictive parameter configurations.
With proven universal approximation capabilities, time-reversible dynamics, and top-tier empirical results, LinOSS sets a new standard for long-range classification, forecasting, and extreme-sequence tasks. Its two specialized variants allow it to adapt across a wide range of systems, striking a balance between stability, scalability, and efficiency.
Journal Reference
Rusch, T. K., & Rus, D. (2024). Oscillatory State-Space Models. arXiv (Cornell University). DOI:10.48550/arxiv.2410.03943. https://arxiv.org/abs/2410.03943
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