neural-ssm 0.30

Neural robust state space models in PyTorch

neural-ssm

PyTorch implementations of state-space models (SSMs) with a built-in robustness certificate in the form of a tunable L2-bound. This is obtained by using:

• free parametrizations of L2-bounded linear dynamical systems
• Lipschitz-bounded static nonlinearities

The mathematical details are in:

• Free Parametrization of L2-bounded State Space Models
  https://arxiv.org/abs/2503.23818

Installation

Install from pip:

pip install neural-ssm

Install the latest GitHub version:

pip install git+https://github.com/LeoMassai/neural-ssm.git

Architecture and robustness recipe

Let's see what is an SSM more in detail

Reading the figure from left to right:

1. Input is projected by an encoder.
2. A stack of SSL blocks is applied.
3. Each block combines:
   • a dynamic core with different state-space parametrizations (lru, l2n, or tv)
   • a static nonlinearity (LGLU, LMLP, GLU, ...)
   • a residual connection.
4. Output is projected by a decoder.

Main message: l2n and tv, when used with a Lipschitz-bounded nonlinearity such as LGLU, enable robust deep SSMs with prescribed L2 bound.

Main parametrizations

• lru: inspired by "Resurrecting Linear Recurrences"; parametrizes all and only stable LTI systems.
• l2n: free parametrization of all and only LTI systems with a prescribed L2 bound.
• tv: free parametrization of a time-varying selective recurrent unit with prescribed L2 bound (paper in preparation).

All these parametrizations support both forward execution modes:

• parallel scan via mode="scan" (tipically very fast for long sequences)
• standard recurrence loop via mode="loop"

You select the mode at call time, e.g. model(u, mode="scan") or model(u, mode="loop").

Main SSM parameters

• d_input: input feature dimension.
• d_output: output feature dimension.
• d_model: latent model dimension used inside each SSL block.
• d_state: internal recurrent state dimension.
• n_layers: number of stacked SSL blocks.
• param: parametrization of the recurrent unit (lru, l2n, tv, ...).
• ff: static nonlinearity type (GLU, MLP, LMLP, LGLU, TLIP).
• gamma: desired L_2 bound of the overall SSM. If gamma=None, it is trainable.

Where each component is in the code

• End-to-end wrapper (encoder, stack, decoder):
  DeepSSM in src/neural_ssm/ssm/lru.py
• Repeated SSM block (dynamic core + nonlinearity + residual):
  SSL in src/neural_ssm/ssm/lru.py
• Dynamic cores:
  • lru -> LRU in src/neural_ssm/ssm/lru.py
  • l2n -> Block2x2DenseL2SSM in src/neural_ssm/ssm/lru.py
  • tv -> RobustMambaDiagSSM in src/neural_ssm/ssm/mamba.py
• Static nonlinearities:
  • GLU, MLP in src/neural_ssm/static_layers/generic_layers.py
  • LGLU, LMLP, TLIP in src/neural_ssm/static_layers/lipschitz_mlps.py
• Parallel scan utilities:
  src/neural_ssm/ssm/scan_utils.py

Quick tutorial

For a complete, runnable training example on a nonlinear benchmark dataset, see: Test_files/Tutorial_DeepSSM.py

Tensor shapes and forward outputs

• Input tensor shape is u: (B, L, d_input) where:
  • B = batch size
  • L = sequence length
  • d_input = input dimension
• Output tensor shape is y: (B, L, d_output).
• DeepSSM returns two objects:
  • y: the model output sequence
  • state: a list of recurrent states (one tensor per SSL block), useful for stateful calls.

State initialization in forward:

• You can pass state= as a list with one initial state tensor for each SSL block.
• If state is not provided (state=None), internal recurrent states are initialized to zero.

How to create and call a Deep SSM

Building and using the SSM is pretty easy:

import torch
from neural_ssm import DeepSSM

model = DeepSSM(
    d_input=1,
    d_output=1,
    d_model=16,
    d_state=16,
    n_layers=4,
    param="tv",
    ff="LGLU",
    gamma=2.0,
)

u = torch.randn(8, 200, 1)               # (B, L, d_input)
y, state = model(u, mode="scan")         # zero-initialized internal states

# Stateful call: pass one state per SSL block
u_next = torch.randn(8, 200, 1)
y_next, state = model(u_next, state=state, mode="scan")

Top-level API

• DeepSSM, SSMConfig
• LRU, L2RU, lruz, PureLRUR, SimpleRNN
• static layers re-exported in neural_ssm.layers

Examples

Example and experiment scripts are available in Test_files/, including:

• Test_files/Tutorial_DeepSSM.py: minimal end-to-end DeepSSM training tutorial.

Citation

If you use this repository in research, please cite:

Free Parametrization of L2-bounded State Space Models
https://arxiv.org/abs/2503.23818