# State space representations of the Roesser type for convolutional layers

arXiv: https://arxiv.org/abs/2403.11938 · HTML: https://mlsanity.com/paper/2403.11938/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 28 references that are in the map.
- First posted on arXiv 18 Mar 2024.

## Builds on (most cited first, 16 of 16)

- [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)
- [A Survey of Convolutional Neural Networks: Analysis, Applications, and Prospects](https://mlsanity.com/paper/2004.02806/) (2020)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [A guide to convolution arithmetic for deep learning](https://mlsanity.com/paper/1603.07285/) (2016)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Lipschitz Bounded Equilibrium Networks](https://mlsanity.com/paper/2010.01732/) (2020)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming](https://mlsanity.com/paper/2004.07876/) (2020)
- [Lipschitz constant estimation for 1D convolutional neural networks](https://mlsanity.com/paper/2211.15253/) (2022)
- [Neural network training under semidefinite constraints](https://mlsanity.com/paper/2201.00632/) (2022)
- [Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramian](https://mlsanity.com/paper/2303.11835/) (2023)
- [Sparse Polynomial Optimisation for Neural Network Verification](https://mlsanity.com/paper/2202.02241/) (2022)
- [Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativity](https://mlsanity.com/paper/2111.12906/) (2021)
- [Convolutional Neural Networks as 2-D systems](https://mlsanity.com/paper/2303.03042/) (2023)

## Built on by (most cited first, 3 of 3)

- [Lipschitz constant estimation for general neural network architectures using control tools](https://mlsanity.com/paper/2405.01125/) (2024)
- [LipKernel: Lipschitz-Bounded Convolutional Neural Networks via Dissipative Layers](https://mlsanity.com/paper/2410.22258/) (2024)
- [Tensor-to-Tensor Models with Fast Iterated Sum Features](https://mlsanity.com/paper/2506.06041/) (2025)

## Abstract (arXiv)

From the perspective of control theory, convolutional layers (of neural networks) are 2-D (or N-D) linear time-invariant dynamical systems. The usual representation of convolutional layers by the convolution kernel corresponds to the representation of a dynamical system by its impulse response. However, many analysis tools from control theory, e.g., involving linear matrix inequalities, require a state space representation. For this reason, we explicitly provide a state space representation of the Roesser type for 2-D convolutional layers with cinr1 + coutr2 states, where cin/cout is the number of input/output channels of the layer and r1/r2 characterizes the width/length of the convolution kernel. This representation is shown to be minimal for cin = cout. We further construct state space representations for dilated, strided, and N-D convolutions.