# Observability conditions for neural state-space models with eigenvalues and their roots of unity

arXiv: https://arxiv.org/abs/2504.15758 · HTML: https://mlsanity.com/paper/2504.15758/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Core SSM architectures for language, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 18 of its 40 references that are in the map.
- First posted on arXiv 22 Apr 2025.

## Placed in

- State-space sequence models (S4, Mamba) / [Core SSM architectures for language](https://mlsanity.com/ssm/core-ssm-architectures-for-language/)

## Builds on (most cited first, 18 of 18)

- [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)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [AdaGrad stepsizes: Sharp convergence over nonconvex landscapes](https://mlsanity.com/paper/1806.01811/) (2018)
- [Lower Bounds for Finding Stationary Points I](https://mlsanity.com/paper/1710.11606/) (2017)
- [Lower Bounds for Finding Stationary Points II: First-Order Methods](https://mlsanity.com/paper/1711.00841/) (2017)
- [Stochastic Gradient Descent for Nonconvex Learning without Bounded Gradient Assumptions](https://mlsanity.com/paper/1902.00908/) (2019)
- [MambaLRP: Explaining Selective State Space Sequence Models](https://mlsanity.com/paper/2406.07592/) (2024)
- [State-space models are accurate and efficient neural operators for dynamical systems](https://mlsanity.com/paper/2409.03231/) (2024)
- [State Space Models as Foundation Models: A Control Theoretic Overview](https://mlsanity.com/paper/2403.16899/) (2024)
- [Functional observability and subspace reconstruction in nonlinear systems](https://mlsanity.com/paper/2301.04108/) (2023)
- [On the Use of the Observability Gramian for Partially Observed Robotic Path Planning Problems](https://mlsanity.com/paper/1801.09877/) (2018)
- [Neural State-Space Models: Empirical Evaluation of Uncertainty Quantification](https://mlsanity.com/paper/2304.06349/) (2023)
- [Observability of complex systems via conserved quantities](https://mlsanity.com/paper/2408.00143/) (2024)
- [Sparse Mamba: Introducing Controllability, Observability, And Stability To Structural State Space Models](https://mlsanity.com/paper/2409.00563/) (2024)

## Abstract (arXiv)

We operate through the lens of ordinary differential equations and control theory to study the concept of observability in the context of neural state-space models and the Mamba architecture. We develop strategies to enforce observability, which are tailored to a learning context, specifically where the hidden states are learnable at initial time, in conjunction to over its continuum, and high-dimensional. We also highlight our methods emphasize eigenvalues, roots of unity, or both. Our methods effectuate computational efficiency when enforcing observability, sometimes at great scale. We formulate observability conditions in machine learning based on classical control theory and discuss their computational complexity. Our nontrivial results are fivefold. We discuss observability through the use of permutations in neural applications with learnable matrices without high precision. We present two results built upon the Fourier transform that effect observability with high probability up to the randomness in the learning. These results are worked with the interplay of representations in Fourier space and their eigenstructure, nonlinear mappings, and the observability matrix. We present a result for Mamba that is similar to a Hautus-type condition, but instead employs an argument using a Vandermonde matrix instead of eigenvectors. Our final result is a shared-parameter construction of the Mamba system, which is computationally efficient in high exponentiation. We develop a training algorithm with this coupling, showing it satisfies a Robbins-Monro condition under certain orthogonality, while a more classical training procedure fails to satisfy a contraction with high Lipschitz constant.