Liquid Structural State-Space Models

Abstract: A proper parametrization of state transition matrices of linear state-space models (SSMs) followed by standard nonlinearities enables them to efficiently learn representations from sequential data, establishing the state-of-the-art on a large series of long-range sequence modeling benchmarks. In this paper, we show that we can improve further when the structural SSM such as S4 is given by a linear liquid time-constant (LTC) state-space model. LTC neural networks are causal continuous-time neural networks with an input-dependent state transition module, which makes them learn to adapt to incoming inputs at inference. We show that by using a diagonal plus low-rank decomposition of the state transition matrix introduced in S4, and a few simplifications, the LTC-based structural state-space model, dubbed Liquid-S4, achieves the new state-of-the-art generalization across sequence modeling tasks with long-term dependencies such as image, text, audio, and medical time-series, with an average performance of 87.32% on the Long-Range Arena benchmark. On the full raw Speech Command recognition, dataset Liquid-S4 achieves 96.78% accuracy with a 30% reduction in parameter counts compared to S4. The additional gain in performance is the direct result of the Liquid-S4's kernel structure that takes into account the similarities of the input sequence samples during training and inference.

• The paper introduces Liquid-S4, which combines LTC networks with S4 to capture long-term dependencies with high expressivity.
• Experimental results show superior performance on benchmarks like Long Range Arena (87.32% average), BIDMC Vital Signs, Sequential CIFAR, and Speech Commands tasks.
• The study demonstrates that integrating liquid kernels with structured state-space architectures advances sequence modeling for applications from medical signals to NLP.

Liquid Structural State-Space Models: A Comprehensive Evaluation

The paper presents an advanced approach to sequence modeling by integrating Liquid Time-Constant (LTC) networks with Structured State-Space Models (S4), resulting in Liquid-S4. This hybrid model aims to enhance the expressivity and generalization capabilities of state-space models in handling long-range dependencies across diverse sequential data, such as image sequences, text, audio, and medical time-series.

S4 and Liquid Time-Constant Networks

Structured State-Space Models, particularly S4, have established themselves as powerful frameworks for sequence modeling. Their effectiveness is largely attributed to the efficient parameterization of their state transition matrices using techniques such as HiPPO (High-order polynomial projection operators) and diagonal plus low-rank decomposition. S4 models have demonstrated superiority over conventional sequence models like RNNs, CNNs, and Transformers, particularly in managing long sequences.

Liquid Time-Constant networks add another layer of sophistication by incorporating input-dependent state transitions. These continuous-time neural networks dynamically adapt to incoming data during inference, thus offering a more nuanced representation of causal dependencies in time-series data. However, their overall complexity and scalability have often been bottlenecked by the requirements related to differential equation solvers.

Liquid-S4: Bridging the Divide

Liquid-S4 capitalizes on the strengths of both S4 and LTC architectures by formulating a linearized LTC model that integrates seamlessly with the S4 structure. This combination results in a model that not only captures long-term dependencies with a high degree of accuracy but also benefits from the adaptability and expressiveness inherent in liquid networks. The introduction of a liquid kernel that considers the covariance among input samples enhances the model's ability to generalize across different domains.

Experimental Evaluation

Through an extensive empirical study, Liquid-S4 is positioned as a leading performer across a number of benchmarks:

• Long Range Arena (LRA) Benchmarks: Liquid-S4 outperforms previous state-of-the-art models, including various S4 variants, in tasks with sequence lengths extending to thousands. With an average performance of 87.32%, it demonstrates significant gains in tasks like ListOps, IMDB, and pixel-level classification on CIFAR.
• BIDMC Vital Signs Dataset: The model excels in predicting heart rate (HR), respiratory rate (RR), and blood oxygen saturation (SpO2), outperforming other sophisticated architectures such as S4-LegS and S4D variants.
• Sequential CIFAR: With sequential data imposed from image grids, Liquid-S4 achieves the highest accuracy among all models tested, demonstrating its capacity to manage complex spatial-temporal dependencies.
• Speech Commands Recognition: For both the full-label set and the reduced ten-class set, the model maintains superior performance, indicating its robustness in audio sequence modeling.

Implications and Future Directions

The results imply significant advancements in sequence modeling capabilities when combining the adaptability of liquid networks with the structured robustness of S4. In practice, Liquid-S4 can be a critical tool for applications that rely on processing long and complex sequences, offering potential improvements in fields ranging from medical signal processing to natural language processing.

Future work could explore further integration of liquid time-constant dynamics with other structured state-space forms or broader application scenarios. Optimizing the computational aspects of liquid kernels remains an essential area of focus, ensuring that the model's complexity does not hinder its application in real-time tasks. Additionally, understanding how Liquid-S4 fares in zero-shot transfer learning scenarios can provide insights into its robustness and flexibility across varied domain shifts.

Overall, Liquid-S4 presents a meaningful step forward in leveraging hybrid neural architectures to advance the state-of-the-art in sequence learning.