# l2 Induced Norm Analysis of Discrete-Time LTI Systems for Nonnegative Input Signals and Its Application to Stability Analysis of Recurrent Neural Networks

arXiv: https://arxiv.org/abs/2011.12726 · HTML: https://mlsanity.com/paper/2011.12726/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Nov 2020.

## Built on by (most cited first, 5 of 5)

- [Stability and Performance Analysis of Discrete-Time ReLU Recurrent Neural Networks](https://mlsanity.com/paper/2405.05236/) (2024)
- [Stability Analysis of Recurrent Neural Networks by IQC with Copositive Mutipliers](https://mlsanity.com/paper/2202.04592/) (2022)
- [Local Lipschitz Constant Computation of ReLU-FNNs: Upper Bound Computation with Exactness Verification](https://mlsanity.com/paper/2310.11104/) (2023)
- [A Unified Representation of Neural Networks Architectures](https://mlsanity.com/paper/2512.17593/) (2025)
- [Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant](https://mlsanity.com/paper/2209.06175/) (2022)

## Abstract (arXiv)

In this paper, we focus on the "positive" l2 induced norm of discrete-time linear time-invariant systems where the input signals are restricted to be nonnegative. To cope with the nonnegativity of the input signals, we employ copositive programming as the mathematical tool for the analysis. Then, by applying an inner approximation to the copositive cone, we derive numerically tractable semidefinite programming problems for the upper and lower bound computation of the "positive" l2 induced norm. This norm is typically useful for the stability analysis of feedback systems constructed from an LTI system and nonlinearities where the nonlinear elements provide only nonnegative signals. As a concrete example, we illustrate the usefulness of the "positive" l2 induced norm for the stability analysis of recurrent neural networks with activation functions being rectified linear units.