# A Novel Schur-Decomposition-Based Weight Projection Method for Stable State-Space Neural-Network Architectures

arXiv: https://arxiv.org/abs/2605.14489 · HTML: https://mlsanity.com/paper/2605.14489/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 47 references that are in the map.
- First posted on arXiv 14 May 2026.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
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- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Identification of block-oriented nonlinear systems starting from linear approximations: A survey](https://mlsanity.com/paper/1607.01217/) (2016)
- [An L-BFGS-B approach for linear and nonlinear system identification under ℓ1 and group-Lasso regularization](https://mlsanity.com/paper/2403.03827/) (2024)
- [Baseline Results for Selected Nonlinear System Identification Benchmarks](https://mlsanity.com/paper/2405.10779/) (2024)
- [On computing the distance to stability for matrices using linear dissipative Hamiltonian systems](https://mlsanity.com/paper/1611.00595/) (2016)
- [Approximating the nearest stable discrete-time system](https://mlsanity.com/paper/1802.08033/) (2018)
- [SIMBa: System Identification Methods leveraging Backpropagation](https://mlsanity.com/paper/2311.13889/) (2023)
- [Nearest Omega-stable matrix via Riemannian optimization](https://mlsanity.com/paper/2002.07052/) (2020)

## Abstract (arXiv)

Building black-box models for dynamical systems from data is a challenging problem in machine learning, especially when asymptotic stability guarantees are required. In this paper, we introduce a novel stability-ensuring and backpropagation-compatible projection scheme based on the Schur decomposition for the state matrix of linear discrete-time state-space layers, as well as an alternative pre-factorized formulation of the methodology. The proposed methods dynamically project the quasi-triangular factor of the state matrix's real Schur decomposition onto its nearest stable peer, ensuring stable dynamics with minimal overparameterization. Experiments on synthetic linear systems demonstrate that, despite a marginal increase in computational complexity, the method achieves accuracy and convergence rates comparable to those of state-of-the-art stable-system identification techniques. Furthermore, the lower weight count facilitates convergence during training without sacrificing accuracy in stacked neural-network architectures with static nonlinearities targeting real-world datasets. These results suggest that the Schur-based projection provides a numerically robust framework for identifying complex dynamics on par with the state of the art while satisfying strict asymptotic-stability requirements.