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Spectral State Space Models (2312.06837v4)

Published 11 Dec 2023 in cs.LG

Abstract: This paper studies sequence modeling for prediction tasks with long range dependencies. We propose a new formulation for state space models (SSMs) based on learning linear dynamical systems with the spectral filtering algorithm (Hazan et al. (2017)). This gives rise to a novel sequence prediction architecture we call a spectral state space model. Spectral state space models have two primary advantages. First, they have provable robustness properties as their performance depends on neither the spectrum of the underlying dynamics nor the dimensionality of the problem. Second, these models are constructed with fixed convolutional filters that do not require learning while still outperforming SSMs in both theory and practice. The resulting models are evaluated on synthetic dynamical systems and long-range prediction tasks of various modalities. These evaluations support the theoretical benefits of spectral filtering for tasks requiring very long range memory.

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Authors (4)
  1. Naman Agarwal (51 papers)
  2. Daniel Suo (11 papers)
  3. Xinyi Chen (79 papers)
  4. Elad Hazan (106 papers)
Citations (9)
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Summary

Spectral State Space Models: Overview and Insights

Spectral State Space Models (SSMs) present a notable advancement in sequence modeling, addressing the critical issue of long-range dependencies in predictive tasks. This paper introduces a novel method leveraging spectral filtering within SSMs, proposing an architecture called Spectral State Space Models. The approach has been validated both theoretically and empirically, showing significant improvements over traditional methods.

Central Contributions

The paper makes several critical contributions to the field of sequence modeling:

  1. Formulation of Spectral State Space Models: The authors propose SSMs based on linear dynamical systems (LDS) optimized through spectral filtering, addressing challenges inherent in recurrent neural networks (RNNs) and transformers.
  2. Theoretical Validation: They provide proof of the robustness and efficiency of spectral filtering, showing its advantages in handling long-range dependencies without being affected by the spectrum of underlying dynamics or problem dimensionality.
  3. Practical Evaluation: Through various synthetic and real-world benchmarks, the paper demonstrates that Spectral SSMs outperform traditional SSMs in both stability and efficiency.

The Problem with Traditional Methods

Handling long-range dependencies in sequence prediction is a well-known challenge. RNNs, while powerful, suffer from vanishing and exploding gradients, making them difficult to train, especially over longer sequences. Transformers, despite their parallelization capabilities and success across domains, have memory and computation requirements that scale quadratically with context length. This limits their efficacy in tasks requiring very long range memory.

Spectral Filtering in SSMs

At the core of the proposed method is spectral filtering, a technique for representing past inputs in a basis that reflects the structure of the system matrices. Unlike traditional linear dynamical systems, whose memory and stability are contingent on the eigenvalues of matrix AA, spectral filtering decouples system memory from these eigenvalues, thus ensuring stability and efficient training even in systems with small δ\delta values (where δ\delta is the distance of eigenvalues from unity).

Spectral Transform Unit (STU)

To implement these concepts, the authors introduce the Spectral Transform Unit (STU), an essential component of their proposed model. The STU applies a fixed set of spectral filters to the input sequence, transforming it into a spectral basis. This reformulation allows for the stable and computationally efficient representation of sequences with long memory:

y^t=y^t−2+∑i=13Miuut+1−i+∑k=1KMkϕ+σk1/4Ut−2,k++∑k=1KMkϕ−σk1/4Ut−2,k−\hat{y}_t = \hat{y}_{t-2} + \sum_{i=1}^{3} M^{u}_{i} u_{t+1-i} + \sum_{k = 1}^K M^{\phi+}_k \sigma_k^{1/4} U^+_{t-2,k} + \sum_{k = 1}^K M^{\phi-}_k \sigma_k^{1/4} U^-_{t-2,k}

Empirical Validation

The authors validate their claims through extensive empirical evaluation. On synthetic datasets simulating marginally stable systems, Spectral SSMs demonstrate faster and more stable training compared to Linear Recurrent Units (LRUs). Further, on the Long Range Arena benchmark, designed to test models on tasks with long-term dependencies, Spectral SSMs show competitive performance across multiple tasks, highlighting their robustness and adaptability.

Broader Implications and Future Work

The development of Spectral SSMs has significant implications for various practical applications requiring long-term memory:

  • Natural Language Processing: In LLMing, where context and dependencies can stretch over long sequences, Spectral SSMs can provide more stable and scalable solutions.
  • Time Series Analysis: For modeling complex time series data, particularly in finance and climatology, where the system dynamics are intricate and long-term dependencies are critical.
  • Audio and Speech Processing: Tasks such as audio generation and speech recognition, which benefit from accurate long-range sequence modeling, stand to gain from the proposed architecture.

Looking forward, the theoretical underpinnings presented in this paper open avenues for further research. Potential directions include extending the spectral filtering technique to asymmetric systems and exploring hybrid models that integrate spectral SSMs with other sequence prediction architectures to further enhance performance.

Conclusion

This paper offers a compelling contribution to sequence modeling by introducing Spectral State Space Models. By leveraging spectral filtering, the authors address fundamental limitations in existing methods, providing a robust, stable, and efficient alternative for handling long-range dependencies. The theoretical and empirical results presented set a new benchmark in the field, paving the way for future advancements in sequence prediction and modeling.

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