Linear Dynamical Systems (LDSs) are fundamental tools for modeling spatiotemporal data in various disciplines.

State-Space Models (SSMs) excel at capturing long-range dependencies with structured recurrence, making them well-suited for sequence modeling. However, their evolving internal states pose challenges in adapting them under Continual Learning (CL). This is particularly difficult in exemplar-free settings, where the absence of prior data leaves updates to the dynamic SSM states unconstrained, resulting in catastrophic forgetting. To address this, we propose Inf-SSM, a novel and simple geometry-aware regularization method that utilizes the geometry of the infinite-dimensional Grassmannian to constrain state evolution during CL. Unlike classical continual learning methods that constrain weight updates, Inf-SSM regularizes the infinite-horizon evolution of SSMs encoded in their extended observability subspace. We show that enforcing this regularization requires solving a matrix equation known as the Sylvester equation, which typically incurs $\mathcal{O}(n^3)$ complexity. We develop a $\mathcal{O}(n^2)$ solution by exploiting the structure and properties of SSMs. This leads to an efficient regularization mechanism that can be seamlessly integrated into existing CL methods. Comprehensive experiments on challenging benchmarks, including ImageNet-R and Caltech-256, demonstrate a significant reduction in forgetting while improving accuracy across sequential tasks.

Linear Dynamical Systems (LDSs) are fundamental tools for modeling spatiotemporal data in various disciplines. Though rich in modeling, analyzing LDSs is not free of difficulty, mainly because LDSs do not comply with Euclidean geometry and hence conventional learning techniques can not be applied directly. In this paper, we propose an efficient projected gradient descent method to minimize a general form of a loss function and demonstrate how clustering and sparse coding with LDSs can be solved by the proposed method efficiently. To this end, we first derive a novel canonical form for representing the parameters of an LDS, and then show how gradient-descent updates through the projection on the space of LDSs can be achieved dexterously. In contrast to previous studies, our solution avoids any approximation in LDS modeling or during the optimization process. Extensive experiments reveal the superior performance of the proposed method in terms of the convergence and classification accuracy over state-of-the-art techniques.

Linear Dynamical Systems (LDSs) are fundamental tools for modeling spatio-temporal data in various disciplines. Though rich in modeling, analyzing LDSs is not free of difficulty, mainly because LDSs do not comply with Euclidean geometry and hence conventional learning techniques can not be applied directly. In this paper, we propose an efficient projected gradient descent method to minimize a general form of a loss function and demonstrate how clustering and sparse coding with LDSs can be solved by the proposed method efficiently. To this end, we first derive a novel canonical form for representing the parameters of an LDS, and then show how gradient-descent updates through the projection on the space of LDSs can be achieved dexterously. In contrast to previous studies, our solution avoids any approximation in LDS modeling or during the optimization process. Extensive experiments reveal the superior performance of the proposed method in terms of the convergence and classification accuracy over state-of-the-art techniques.