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Structured State Spaces for Sequence Modeling

This repository provides the official implementations and experiments for models related to S4, including HiPPO, LSSL, SaShiMi, DSS, HTTYH, S4D, and S4ND.

Project-specific information for each of these models, including overview of the source code and specific experiment reproductions, can be found under models/.

Table of Contents

Setting up the environment and porting S4 to external codebases:

Using this repository for training models:




  • More documentation for training from scratch using this repository
  • Compilation of S4 resources and implementations
  • pip package



This repository requires Python 3.9+ and Pytorch 1.10+. It has been tested up to Pytorch 1.13.1. Other packages are listed in requirements.txt. Some care may be needed to make some of the library versions compatible, particularly torch/torchvision/torchaudio/torchtext.

Example installation:

conda install pytorch==1.13.1 torchvision==0.14.1 torchaudio==0.13.1 pytorch-cuda=11.6 -c pytorch -c nvidia
pip install -r requirements.txt

Structured Kernels

A core operation of S4 are the Cauchy and Vandermonde kernels described in the paper. These are very simple matrix multiplications; a naive implementation of these operation can be found in the standalone in the function cauchy_naive and log_vandermonde_naive. However, as the paper describes, this has suboptimal memory usage that currently requires a custom kernel to overcome in PyTorch.

Two more efficient methods are supported. The code will automatically detect if either of these is installed and call the appropriate kernel.

Custom CUDA Kernel

This version is faster but requires manual compilation for each machine environment. Run python install from the directory extensions/kernels/.


This version is provided by the pykeops library. Installation usually works out of the box with pip install pykeops cmake which are also listed in the requirements file.

Getting Started with S4

S4 Module

Self-contained files for the S4 layer and variants can be found in models/s4/, which includes instructions for calling the module.

See notebooks/ for visualizations explaining some concepts behind HiPPO and S4.

Example Train Script (External Usage) is a self-contained training script for MNIST and CIFAR that imports the standalone S4 file. The default settings python reaches 88% accuracy on sequential CIFAR with a very simple S4D model of 200k parameters. This script can be used as an example for using S4 variants in external repositories.

Training with this Repository (Internal Usage)

This repository aims to provide a very flexible framework for training sequence models. Many models and datasets are supported.

The basic entrypoint is python -m train, or equivalently

python -m train pipeline=mnist model=s4

which trains an S4 model on the Permuted MNIST dataset. This should get to around 90% after 1 epoch which takes 1-3 minutes depending on GPU.

More examples of using this repository are documented throughout. See Training for an overview.

Optimizer Hyperparameters

One important feature of this codebase is supporting parameters that require different optimizer hyperparameters. In particular, the SSM kernel is particularly sensitive to the $(A, B)$ (and sometimes $\Delta$ parameters), so the learning rate on these parameters is sometimes lowered and the weight decay is always set to $0$.

See the method register in the model (e.g. and the function setup_optimizer in the training script (e.g. for an examples of how to implement this in external repos.


The core training infrastructure of this repository is based on Pytorch-Lightning with a configuration scheme based on Hydra.

The main entrypoint is and configs are found in configs/.


Basic datasets are auto-downloaded, including MNIST, CIFAR, and Speech Commands. All logic for creating and loading datasets is in src/dataloaders directory. The README inside this subdirectory documents how to download and organize other datasets.


Models are defined in src/models. See the README in this subdirectory for an overview.

Configs and Hyperparameters

Pre-defined configs reproducing end-to-end experiments from the papers are provided, found under project-specific information in models/, such as for the original S4 paper.

Configs can also be easily modified through the command line. An example experiment is

python -m train pipeline=mnist dataset.permute=True model=s4 model.n_layers=3 model.d_model=128 model.norm=batch model.prenorm=True wandb=null

This uses the Permuted MNIST task with an S4 model with a specified number of layers, backbone dimension, and normalization type.

See configs/ for more detailed documentation about the configs.


It is recommended to read the Hydra documentation to fully understand the configuration framework. For help launching specific experiments, please file an issue.


Each experiment will be logged to its own directory (generated by Hydra) of the form ./outputs/<date>/<time>/. Checkpoints will be saved here inside this folder and printed to console whenever a new checkpoint is created. To resume training, simply point to the desired .ckpt file (a PyTorch Lightning checkpoint, e.g. ./outputs/<date>/<time>/checkpoints/val/loss.ckpt) and append the flag train.ckpt=<path>/<to>/<checkpoint>.ckpt to the original training command.

PyTorch Lightning Trainer

The PTL Trainer class controls the overall training loop and also provides many useful pre-defined flags. Some useful examples are explained below. The full list of allowable flags can be found in the PTL documentation, as well as our trainer configs. See the default trainer config configs/trainer/default.yaml for the most useful options.

Multi-GPU training

Simply pass in trainer.gpus=2 to train with 2 GPUs.

Inspect model layers

trainer.weights_summary=full prints out every layer of the model with their parameter counts. Useful for debugging internals of models.

Data subsampling

trainer.limit_{train,val}_batches={10,0.1} trains (validates) on only 10 batches (0.1 fraction of all batches). Useful for testing the train loop without going through all the data.


Logging with WandB is built into this repository. In order to use this, simply set your WANDB_API_KEY environment variable, and change the wandb.project attribute of configs/config.yaml (or pass it on the command line e.g. python -m train .... wandb.project=s4).

Set wandb=null to turn off WandB logging.


Autoregressive generation can be performed with the script. This script can be used in two ways after training a model using this codebase.

Option 1: Checkpoint Path

The more flexible option requires the checkpoint path of the trained PyTorch Lightning model. The generation script accepts the same config options as the train script, with a few additional flags that are documented in configs/generate.yaml. After training with python -m train <train flags>, generate with

python -m generate <train flags> checkpoint_path=<path/to/model.ckpt> <generation flags>

Any of the flags found in the config can be overridden.

Note: This option can be used with either .ckpt checkpoints (PyTorch Lightning, which includes information for the Trainer) or .pt checkpoints (PyTorch, which is just a model state dict).

Option 2: Experiment Path

The second option for generation does not require passing in training flags again, and instead reads the config from the Hydra experiment folder, along with a PyTorch Lightning checkpoint within the experiment folder.

Example 1 (Language)

Download the WikiText-103 model checkpoint, for example to ./checkpoints/ This model was trained with the command python -m train experiment=lm/s4-wt103. Note that from the config we can see that the model was trained with a receptive field of length 8192.

To generate, run

python -m generate experiment=lm/s4-wt103 checkpoint_path=checkpoints/ n_samples=1 l_sample=16384 l_prefix=8192 decode=text

This generates a sample of length 16384 conditioned on a prefix of length 8192.

Example 2 (Audio)

Let's train a small SaShiMi model on the SC09 dataset. We can also reduce the number of training and validation batches to get a checkpoint faster:

python -m train experiment=audio/sashimi-sc09 model.n_layers=2 trainer.limit_train_batches=0.1 trainer.limit_val_batches=0.1

After the first epoch completes, a message is printed indicating where the checkpoint is saved.

Epoch 0, global step 96: val/loss reached 3.71754 (best 3.71754), saving model to "<repository>/outputs/<date>/<time>/checkpoints/val/loss.ckpt"

Option 1:

python -m generate experiment=audio/sashimi-sc09 model.n_layers=2 checkpoint_path=<repository>/outputs/<date>/<time>/checkpoints/val/loss.ckpt n_samples=4 l_sample=16000

This option redefines the full config so that the model and dataset can be constructed.

Option 2:

python -m generate experiment_path=<repository>/outputs/<date>/<time> checkpoint_path=checkpoints/val/loss.ckpt n_samples=4 l_sample=16000

This option only needs the path to the Hydra experiment folder and the desired checkpoint within.

Overall Repository Structure

configs/         Config files for model, data pipeline, training loop, etc.
data/            Default location of raw data
extensions/      CUDA extensions (Cauchy and Vandermonde kernels)
src/             Main source code for models, datasets, etc.
  callbacks/     Training loop utilities (e.g. checkpointing)
  dataloaders/   Dataset and dataloader definitions
  models/        Model definitions
  tasks/         Encoder/decoder modules to interface between data and model backbone
models/          Model-specific information (code, experiments, additional resources)       Example training script for using S4 externally         Training entrypoint for this repo      Autoregressive generation script


If you use this codebase, or otherwise found our work valuable, please cite S4 and other relevant papers.

  title={Efficiently Modeling Long Sequences with Structured State Spaces},
  author={Gu, Albert and Goel, Karan and R\'e, Christopher},
  booktitle={The International Conference on Learning Representations ({ICLR})},