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

2025-04-22

AI Summary

This paper addresses the observability guarantee problem for neural state-space models—including Mamba architectures—in high-dimensional, continuous-time, and learnable hidden-state settings. We propose a systematic methodology grounded in control theory and Fourier analysis. First, we generalize the Hautus test to a Vandermonde-matrix formulation, thereby characterizing observability via eigenvalue and unit-root structures. Second, we introduce a high-probability observability construction strategy based on Fourier transforms. Third, we design a shared-parameter Mamba system coupled with a stochastic training algorithm satisfying the Robbins–Monro conditions. Our theoretical contributions comprise five nontrivial results: (i) permutation robustness of observability; (ii) Fourier-domain observability guarantees; (iii) a Mamba-specific observability criterion; (iv) convergence proof of the proposed algorithm; and (v) an efficient computational implementation for large-scale observability constraints.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsIntelligent Robots: State EstimationCognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systemsSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• Study observability in neural state-space models using control theory
• Develop strategies to enforce observability in high-dimensional learning contexts
• Formulate computationally efficient observability conditions for machine learning

Innovation

• Enforce observability via eigenvalues and roots
• Use Fourier transform for high-probability observability
• Shared-parameter Mamba system for efficiency