arXiv subjects

Gustave Bainier

Publications and source records attributed to Gustave Bainier.

5 recordsLinked to original sources

arXiv2026

State-space fading memory

The fading-memory (FM) property captures the progressive loss of influence of past inputs on a system's current output and has originally been formalized by Boyd and Chua in an operator-theoretic framework. Despite its importance for systems approximation, reservoir computing, and recurrent neural networks, its connection with state-space notions of nonlinear stability, especially incremental ones, remains understudied. This paper introduces a state-space definition of FM. In state-space, FM can be interpreted as an extension of incremental input-to-output stability ($\delta$IOS) that explicitly incorporates a memory kernel upper-bounding the decay of past input differences. It is also closely related to Boyd and Chua's FM definition, with the sole difference of requiring uniform, instead of general, continuity of the memory functional with respect to an input-fading norm. We demonstrate that incremental input-to-state stability ($\delta$ISS) implies FM semi-globally for time-invariant systems under an equibounded input assumption. Notably, Boyd and Chua's approximation theorems apply to $\delta$ISS state-space models. As a closing application, we show that, under mild assumptions, the state-space model of a current-driven memristive device possesses the FM property.

eess.SY↗

arXiv2026

Context-dependent manifold learning: A neuromodulated constrained autoencoder approach

Many physical systems exhibit a low-dimensional structure that varies with external parameters: link lengths in a robot, forcing constants in a fluid, or Reynolds numbers in a flow shift the underlying manifold while preserving its intrinsic dimension. Constrained AutoEncoders (cAEs) learn such manifolds through an idempotent encoder-decoder projection, a property that unconstrained autoencoders cannot match and that is essential whenever the model is applied iteratively. However, the standard strategies for making a cAE context-dependent, namely concatenating the context to the input or affinely modulating hidden activations, break the encoder-decoder idempotency, sacrificing the projection guarantee precisely in the setting where it would be most valuable. To restore this guarantee under context variation, we developed the Neuromodulated Constrained Autoencoder (NcAE), which modulates the activation slope and bias of a cAE through a context-driven hyper-network. This paper presents the NcAE, its theoretical foundation, and its empirical validation. We prove that for every context, including contexts unseen at training time, the reconstruction map remains an idempotent projection, the topology of the learned manifold is invariant, and context perturbations induce smooth changes in the manifold. We evaluated our approach on a 16-DoF pendulum with context-dependent coupling and the Lorenz96 system across a bifurcation. The NcAE matched or exceeded the best of six baselines on reconstruction, idempotency, and latent-geometry metrics, while being the only architecture that preserves geometric consistency by construction. The NcAE thereby provides a stable, geometry-preserving coordinate system across families of physical regimes.

cs.LG↗

arXiv2025

From Consensus to Robust Clustering: Multi-Agent Systems with Nonlinear Interactions

This paper establishes a theoretical framework to describe the transition from consensus to stable clustering in multi-agent systems with nonlinear, cooperative interactions. We first establish a sharp threshold for consensus. For a broad class of non-decreasing, Lipschitz-continuous interactions, an explicit inequality linking the interaction's Lipschitz constant to the second-largest eigenvalue of the normalized adjacency matrix of the interaction graph confines all system equilibria to the synchronization manifold. This condition is shown to be a sharp threshold, as its violation permits the emergence of non-synchronized equilibria. We also demonstrate that such clustered states can only arise if the interaction law itself possesses specific structural properties, such as unstable fixed points. For the clustered states that emerge, we introduce a formal framework using Input-to-State Stability (ISS) theory to quantify their robustness. This approach allows us to prove that the internal cohesion of a cluster is robust to perturbations from the rest of the network. The analysis reveals a fundamental principle: cluster coherence is limited not by the magnitude of external influence, but by its heterogeneity across internal nodes. This unified framework, explaining both the sharp breakdown of consensus and the quantifiable robustness of the resulting modular structures, is validated on Zachary's Karate Club network, used as a classic benchmark for community structure.

eess.SY↗

arXiv2025

Directional excitability in Hilbert spaces

We introduce a generalized excitable system in which spikes can happen in a continuum of directions, therefore drastically enriching the expressivity and control capability of the spiking dynamics. In this generalized excitable system, spiking trajectories happen in a Hilbert space with an excitable resting state at the origin and spike responses that can be triggered in any direction as a function of the system's state and inputs. State-dependence of the spiking direction provide the system with a vanishing spiking memory trace, which enables robust tracking and integration of inputs in the spiking direction history. The model exhibits generalized forms of both Hodgkin's Type I and Type II excitability, capturing their usual bifurcation behaviors in an abstract setting. When used as the controller of a two-dimensional navigation task, this model facilitates both the sparseness of the actuation and its sensitivity to environmental inputs. These results highlight the potential of the proposed generalized excitable model for excitable control in high- and infinite-dimensional spaces.

eess.SY↗

arXiv2024

Orthogonal Projection of Convex Sets with a Differentiable Boundary

Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior) and the boundary of its orthogonal projection onto the linear subspaces of the Euclidean space. A system of equations for these orthogonal projections is derived from this topological link. This result is illustrated by the projection of the unit ball of norm $4$ in $\mathbb{R}^3$ on a plane.

math.DG↗