Toeplitz Neural Network for Sequence Modeling (2305.04749v1)

Abstract: Sequence modeling has important applications in natural language processing and computer vision. Recently, the transformer-based models have shown strong performance on various sequence modeling tasks, which rely on attention to capture pairwise token relations, and position embedding to inject positional information. While showing good performance, the transformer models are inefficient to scale to long input sequences, mainly due to the quadratic space-time complexity of attention. To overcome this inefficiency, we propose to model sequences with a relative position encoded Toeplitz matrix and use a Toeplitz matrix-vector production trick to reduce the space-time complexity of the sequence modeling to log linear. A lightweight sub-network called relative position encoder is proposed to generate relative position coefficients with a fixed budget of parameters, enabling the proposed Toeplitz neural network to deal with varying sequence lengths. In addition, despite being trained on 512-token sequences, our model can extrapolate input sequence length up to 14K tokens in inference with consistent performance. Extensive experiments on autoregressive and bidirectional LLMing, image modeling, and the challenging Long-Range Arena benchmark show that our method achieves better performance than its competitors in most downstream tasks while being significantly faster. The code is available at https://github.com/OpenNLPLab/Tnn.

• The paper presents TNN, which leverages a Toeplitz matrix to reduce quadratic complexity to O(n log n) while maintaining high performance.
• It incorporates a lightweight relative position encoder and an exponential decay bias that enables effective extrapolation to sequences up to 14K tokens.
• Empirical results across language and image tasks demonstrate TNN’s scalability and competitive accuracy compared to state-of-the-art models.

Toeplitz Neural Network for Sequence Modeling

The paper introduces the Toeplitz Neural Network (TNN), a novel architecture for efficient sequence modeling, capitalizing on relative positional information while circumventing the computational intensity inherent in conventional transformer models. This approach addresses critical challenges in handling long sequences across domains such as natural language processing and computer vision.

Core Contributions

The central innovation is the use of a Toeplitz matrix, which substantially reduces the quadratic space-time complexity typical in transformers to a log-linear complexity. Unlike traditional transformers that utilize attention mechanisms involving pairwise token relations and positional embedding, the TNN leverages a relative position encoded Toeplitz matrix. The transformation captures token interactions efficiently, reducing computational burden without sacrificing performance.

A key advantage of the Toeplitz structure is its ability to represent relationships with significantly fewer parameters and perform matrix-vector operations in $O(n\log n)$ time. This is achieved through a specialized Toeplitz matrix-vector product trick, which is computationally attractive for long sequence modeling tasks.

Relative Position Encoder and Exponential Decay Bias

To endow the model with the ability to handle varying sequence lengths without parameter expansion, a lightweight relative position encoder generates appropriate positional coefficients. This encoder decouples parameter count from sequence length and allows the network to maintain performance even when facing sequences longer than those seen during training.

For seamless sequence extrapolation, the authors propose an exponential decay bias applied to the Toeplitz matrix. This bias mechanism enables TNN to extend its capacity to considerably longer sequences, up to 14K tokens from a training maximum of 512 tokens, which is a non-trivial enhancement over existing architectures.

Empirical Validation

The TNN is validated through extensive experiments across various benchmarks:

• Autoregressive and Bidirectional LLMing: The model demonstrates competitive or superior perplexity scores compared to state-of-the-art models, affirming its efficacy in natural language tasks.
• Long-Range Arena Benchmark: TNN significantly outperforms competitors on tasks that stress-test the ability to model long-range dependencies, highlighting its robustness and efficiency.
• Image Modeling: Implemented within a visual transformer framework, TNN sustains comparable accuracy on image classification tasks, thereby underscoring its versatility across modalities.

Theoretical and Practical Implications

Theoretically, TNN presents a unified approach to sequence modeling that encapsulates transformers, CNNs, and state-space models as special cases. This broader perspective could pave the way for further research into generalized architectures that efficiently balance complexity and capacity.

Practically, the reduced computational demand and enhanced capacity to generalize over longer sequences hold promise for deploying models in resource-constrained environments, such as edge devices or low-latency applications.

Future Directions

As research into sequence modeling continues to evolve, potential areas of exploration include:

• Optimization of Relative Position Encoding: Further exploration of the parameterization in the relative position encoder to enhance adaptability and efficiency.
• Integration with Advanced Attention Mechanisms: Seeking synergies between Toeplitz-based approaches and emerging efficient attention variants.
• Cross-Domain Applications: Expanding application beyond NLP and vision, potentially into areas such as genomics or complex systems simulation, where sequence modeling plays a critical role.

In conclusion, the Toeplitz Neural Network offers a computationally efficient, scalable solution for sequence modeling, with implications that extend into theoretical unification and practical deployment across various domains.