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[2505.11602] Regularity and Stability Properties of Selective SSMs with Discontinuous Gating
Summary
This paper explores the regularity and stability properties of selective state-space models (SSMs) with discontinuous gating, focusing on their stability analysis and implications for machine learning applications.
Why It Matters
Understanding the stability of selective SSMs is crucial for developing robust machine learning models that can handle discontinuous inputs. This research provides insights into the conditions necessary for stability and the implications for model design, which are vital for practitioners in machine learning and control systems.
Key Takeaways
- Selective SSMs can achieve exponential decay of trajectories under specific conditions.
- Quadratic storage functions are necessary for ensuring passivity in systems with discontinuous gating.
- The study introduces conditions for global Input-to-State Stability (ISS) in selective SSMs.
Computer Science > Machine Learning arXiv:2505.11602 (cs) [Submitted on 16 May 2025 (v1), last revised 24 Feb 2026 (this version, v2)] Title:Regularity and Stability Properties of Selective SSMs with Discontinuous Gating Authors:Nikola Zubić, Davide Scaramuzza View a PDF of the paper titled Regularity and Stability Properties of Selective SSMs with Discontinuous Gating, by Nikola Zubi\'c and 1 other authors View PDF HTML (experimental) Abstract:Deep selective State-Space Models (SSMs), whose state-space parameters are modulated online by a selection signal, offer significant expressive power but pose challenges for stability analysis, especially under discontinuous gating. We study continuous-time selective SSMs through the lenses of passivity and Input-to-State Stability (ISS), explicitly distinguishing the selection schedule $x(\cdot)$ from the driving (port) input $u(\cdot)$. First, we show that state-strict dissipativity ($\beta>0$) together with quadratic bounds on a storage functional implies exponential decay of homogeneous trajectories ($u\equiv 0$), yielding exponential forgetting. Second, by freezing the selection ($x(t)\equiv 0$) we obtain a passive LTV input-output subsystem and prove that its minimal available storage is necessarily quadratic, $V_{a,0}(t,h)=\tfrac{1}{2}h^H Q_0(t)h,$ with $Q_0 \in \mathrm{AUC}_{\mathrm{loc}}$, accommodating discontinuities induced by gating. Third, under the strong hypothesis that a single quadratic storage certifies passivity ...