# LipSSM: Structurally Lipschitz-Bounded Cascaded State-Space Model via Metric Transfer between Consecutive SSM Layers

arXiv: https://arxiv.org/abs/2609.30973 · HTML: https://mlsanity.com/paper/2609.30973/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 30 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 25 of 25)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Hungry Hungry Hippos: Towards Language Modeling with State Space Models](https://mlsanity.com/paper/2212.14052/) (2022)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [Orthogonal Recurrent Neural Networks with Scaled Cayley Transform](https://mlsanity.com/paper/1707.09520/) (2017)
- [A Dynamical System Perspective for Lipschitz Neural Networks](https://mlsanity.com/paper/2110.12690/) (2021)
- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Almost-Orthogonal Layers for Efficient General-Purpose Lipschitz Networks](https://mlsanity.com/paper/2208.03160/) (2022)
- [1-Lipschitz Layers Compared: Memory, Speed, and Certifiable Robustness](https://mlsanity.com/paper/2311.16833/) (2023)
- [Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramian](https://mlsanity.com/paper/2303.11835/) (2023)
- [Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equations](https://mlsanity.com/paper/2304.02976/) (2023)
- [Lipschitz constant estimation for general neural network architectures using control tools](https://mlsanity.com/paper/2405.01125/) (2024)
- [Unconstrained learning of networked nonlinear systems via free parametrization of stable interconnected operators](https://mlsanity.com/paper/2311.13967/) (2023)
- [LipKernel: Lipschitz-Bounded Convolutional Neural Networks via Dissipative Layers](https://mlsanity.com/paper/2410.22258/) (2024)
- [R2DN: Scalable Parameterization of Contracting and Lipschitz Recurrent Deep Networks](https://mlsanity.com/paper/2504.01250/) (2025)
- [Incremental Stability and Performance Analysis of Discrete-Time Nonlinear Systems using the LPV Framework](https://mlsanity.com/paper/2103.10819/) (2021)
- [LipsAM: Lipschitz-Continuous Amplitude Modifier for Audio Signal Processing and its Application to Plug-and-Play Dereverberation](https://mlsanity.com/paper/2603.21684/) (2026)
- [LipsAM: Lipschitz-continuous Neural Networks for Convergent Plug-and-Play Audio Signal Recovery](https://mlsanity.com/paper/2608.23038/) (2026)

## Abstract (arXiv)

Lipschitz continuity is a fundamental principle in the design of certifiably robust deep neural networks (DNNs), wherein adjusting the Lipschitz constant, which quantifies network robustness, is of central theoretical importance. A standard approach to enforcing Lipschitz continuity requires each layer of a DNN to be Lipschitz continuous, thereby guaranteeing overall Lipschitz continuity. However, this layer-wise approach typically imposes overly conservative restrictions by producing a loose estimate of the overall Lipschitz constant, which limits the expressive capacity of the DNN and degrades empirical performance at a prescribed level of robustness. To overcome this loose estimation, the recently proposed LipKernel transfers information across layers to yield a much tighter overall Lipschitz bound than conventional layer-wise construction. In this paper, we extend this concept to cascaded state-space models (SSMs) to construct Lipschitz-continuous DNNs capable of modeling longer-term dependencies. The proposed architecture, named LipSSM, is theoretically justified and empirically evaluated.