Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers (2110.13985v1)

Abstract: Recurrent neural networks (RNNs), temporal convolutions, and neural differential equations (NDEs) are popular families of deep learning models for time-series data, each with unique strengths and tradeoffs in modeling power and computational efficiency. We introduce a simple sequence model inspired by control systems that generalizes these approaches while addressing their shortcomings. The Linear State-Space Layer (LSSL) maps a sequence $u \mapsto y$ by simply simulating a linear continuous-time state-space representation $\dot{x} = Ax + Bu, y = Cx + Du$. Theoretically, we show that LSSL models are closely related to the three aforementioned families of models and inherit their strengths. For example, they generalize convolutions to continuous-time, explain common RNN heuristics, and share features of NDEs such as time-scale adaptation. We then incorporate and generalize recent theory on continuous-time memorization to introduce a trainable subset of structured matrices $A$ that endow LSSLs with long-range memory. Empirically, stacking LSSL layers into a simple deep neural network obtains state-of-the-art results across time series benchmarks for long dependencies in sequential image classification, real-world healthcare regression tasks, and speech. On a difficult speech classification task with length-16000 sequences, LSSL outperforms prior approaches by 24 accuracy points, and even outperforms baselines that use hand-crafted features on 100x shorter sequences.

• The paper presents the LSSL framework that merges recurrent, convolutional, and continuous-time models to capture long-term dependencies.
• It leverages structured matrix computations and continuous state-space representations to generalize convolutional kernels and recurrent architectures.
• Empirical results demonstrate that LSSL outperforms benchmarks in tasks like sequential image classification and speech processing, highlighting its scalability.

Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers

The paper introduces a novel sequence model that combines the strengths of recurrent neural networks (RNNs), convolutional neural networks (CNNs), and neural differential equations (NDEs) through Linear State-Space Layers (LSSL). This approach is aimed at addressing limitations in modeling long sequences while leveraging computational efficiency and training scalability.

Main Contributions

LSSL Framework

The LSSL maps a sequence by simulating a linear continuous-time state-space representation. This abstraction translates into mapping inputs via

• Recurrent Properties: LSSL can be discretized into a linear recurrence, allowing stateful inference with fixed memory usage.
• Convolutional Attributes: The layer represents a convolution, improving parallelizability during training.
• Continuous-time Features: As an implicit differential equation, LSSL adapts across different time scales and can handle irregularly spaced data.

Surprisingly, LSSL not only integrates these paradigms but also generalizes popular convolutional and recurrent models.

Theoretical Insights

The paper reveals underlying connections between LSSLs, RNNs, and CNNs:

• Convolutional Equivalence: All one-dimensional convolutional kernels can be approximated by an LSSL, taking advantage of state-space forms that resemble continuous convolution.
• RNN Generalization: LSSLs are shown to correspond to RNN structures, where gating mechanisms emerge naturally from the discretization of ODEs, offering insight into architectural heuristics traditionally used in RNN design.

Handling Long Dependencies

To overcome limitations inherent in CNNs and RNNs regarding long-term dependencies, LSSLs incorporate structured matrices derived from the HiPPO framework. This approach supports continuous-time memory representations, providing a trainable foundation that captures extended temporal dependencies.

Empirical Validation

Empirically, LSSL outperforms existing sequence models across several benchmarks for varied time series tasks, including sequential image classification and complex speech data challenges, demonstrating its efficacy for sequences as long as 38,000 time steps. These results underscore the model's potential in long-sequence applications, surpassing handcrafted feature baselines in specific tasks.

Computational Considerations

By employing structured matrix computations, LSSLs achieve a degree of computational efficiency, particularly through scalable algorithms for matrix operations. While offering significant improvements over naive implementations, the structured matrices facilitate feasible training and inference workloads.

Implications and Future Work

The integration of state-space representations with deep learning techniques in LSSL promises a unified framework that extends across data modalities, bridging the gap between theory and application for sequence data. The potential to refine computational strategies further, particularly in handling structured matrices, indicates a fertile direction for future research. This could include exploration into more efficient, stable numerical algorithms or extending LSSL concepts to a broader class of sequence modeling challenges.

In summary, this work provides a comprehensive framework by merging traditional and novel deep learning approaches, presenting both theoretical insights and substantial empirical advancements in sequence modeling.