# Efficient State Space Model via Fast Tensor Convolution and Block Diagonalization

arXiv: https://arxiv.org/abs/2402.15290 · HTML: https://mlsanity.com/paper/2402.15290/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Core SSM architectures for language, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 49 of its 72 references that are in the map.
- First posted on arXiv 23 Feb 2024.

## Placed in

- State-space sequence models (S4, Mamba) / [Core SSM architectures for language](https://mlsanity.com/ssm/core-ssm-architectures-for-language/)

## Builds on (most cited first, 30 of 49)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Longformer: The Long-Document Transformer](https://mlsanity.com/paper/2004.05150/) (2020)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Big Bird: Transformers for Longer Sequences](https://mlsanity.com/paper/2007.14062/) (2020)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Generating Long Sequences with Sparse Transformers](https://mlsanity.com/paper/1904.10509/) (2019)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [Long Range Arena: A Benchmark for Efficient Transformers](https://mlsanity.com/paper/2011.04006/) (2020)
- [Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers](https://mlsanity.com/paper/2110.13985/) (2021)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Nystr"omformer: A Nystr"om-Based Algorithm for Approximating Self-Attention](https://mlsanity.com/paper/2102.03902/) (2021)
- [On the Parameterization and Initialization of Diagonal State Space Models](https://mlsanity.com/paper/2206.11893/) (2022)
- [Diagonal State Spaces are as Effective as Structured State Spaces](https://mlsanity.com/paper/2203.14343/) (2022)
- [Long Range Language Modeling via Gated State Spaces](https://mlsanity.com/paper/2206.13947/) (2022)
- [Adversarial Audio Synthesis](https://mlsanity.com/paper/1802.04208/) (2018)
- [Scene Text Recognition with Permuted Autoregressive Sequence Models](https://mlsanity.com/paper/2207.06966/) (2022)
- [Independently Recurrent Neural Network (IndRNN): Building A Longer and Deeper RNN](https://mlsanity.com/paper/1803.04831/) (2018)
- [Liquid Structural State-Space Models](https://mlsanity.com/paper/2209.12951/) (2022)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [ListOps: A Diagnostic Dataset for Latent Tree Learning](https://mlsanity.com/paper/1804.06028/) (2018)
- [Fast Transformers with Clustered Attention](https://mlsanity.com/paper/2007.04825/) (2020)

## Abstract (arXiv)

Existing models encounter bottlenecks in balancing performance and computational efficiency when modeling long sequences. Although the state space model (SSM) has achieved remarkable success in handling long sequence tasks, it still faces the problem of large number of parameters. In order to further improve the efficiency of SSM, we propose a new state space layer based on multiple-input multiple-output SSM, called efficient SSM (eSSM). Our eSSM is built on the convolutional representation of multi-input and multi-input (MIMO) SSM. We propose a variety of effective strategies to improve the computational efficiency. The diagonalization of the system matrix first decouples the original system. Then a fast tensor convolution is proposed based on the fast Fourier transform. In addition, the block diagonalization of the SSM further reduces the model parameters and improves the model flexibility. Extensive experimental results show that the performance of the proposed model on multiple databases matches the performance of state-of-the-art models, such as S4, and is significantly better than Transformers and LSTM. In the model efficiency benchmark, the parameters of eSSM are only 12.89% of LSTM and 13.24% of Mamba. The training speed of eSSM is 3.94 times faster than LSTM and 1.35 times faster than Mamba. Code is available at: hrefhttps://github.com/leonty1/essmhttps://github.com/leonty1/essm.