Regularity and Stability Properties of Selective SSMs with Discontinuous Gating
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911463463976960 |
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| author | Zubić, Nikola Scaramuzza, Davide |
| author_facet | Zubić, Nikola Scaramuzza, Davide |
| contents | 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 ($β>0$) 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 uniformly over all admissible selection schedules, we derive a parametric LMI and universal kernel constraints on gating, formalizing an "irreversible forgetting" structure. Finally, we give sufficient conditions for global ISS with respect to the port input $u(\cdot)$, uniformly over admissible selection schedules, and we validate the main predictions in targeted simulation studies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity and Stability Properties of Selective SSMs with Discontinuous Gating Zubić, Nikola Scaramuzza, Davide Machine Learning Dynamical Systems Optimization and Control 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 ($β>0$) 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 uniformly over all admissible selection schedules, we derive a parametric LMI and universal kernel constraints on gating, formalizing an "irreversible forgetting" structure. Finally, we give sufficient conditions for global ISS with respect to the port input $u(\cdot)$, uniformly over admissible selection schedules, and we validate the main predictions in targeted simulation studies. |
| title | Regularity and Stability Properties of Selective SSMs with Discontinuous Gating |
| topic | Machine Learning Dynamical Systems Optimization and Control |
| url | https://arxiv.org/abs/2505.11602 |