Kazuhiro Sato

Tobias Damm, Kazuhiro Sato, Axel Vierling

We suggest and compare different methods for the numerical solution of Lyapunov like equations with application to control of Markovian jump linear systems. First, we consider fixed point iterations and associated Krylov subspace formulations. Second, we reformulate the equation as an optimization problem and consider steepest descent, conjugate gradient, and a trust-region method. Numerical experiments illustrate that for large-scale problems the trust-region method is more effective than the steepest descent and the conjugate gradient methods. The fixed-point approach, however, is superior to the optimization methods. As an application we consider a networked control system, where the Markov jumps are induced by the wireless communication protocol.

Hiroki Sakamoto, Kazuhiro Sato

This paper develops a data-driven time-limited h2 model reduction method for discrete-time linear time-invariant systems. Specifically, we formulate and solve a regularized time-limited h2 model reduction problem using only noisy impulse response data. Furthermore, we show that the objective function and its gradient can be represented using only noisy impulse response data. Numerical experiments using SLICOT benchmarks demonstrate that the proposed regularized method achieves lower relative time-limited h2 errors than the tested alternatives and is effective in situations where the unregularized method may deteriorate under noise.

Ryoji Anzaki, Kazuhiro Sato

We propose a system identification method, Non-Markovian Optimization-based Modeling for Approximate Dynamics with Spatially-homogeneous memory (NOMADS), for identifying linear dynamical systems from a set of multi-dimensional time-series data obtained through multiple partially excited experiments. NOMADS formulates model identification as a convex optimization problem, in which the state-space coefficient matrices and a memory kernel are estimated jointly under physically motivated constraints using projected gradient descent. The proposed framework models memory effects through a spatially homogeneous kernel, enabling scalable identification of non-Markovian dynamics while keeping the number of free parameters moderate. This structure allows NOMADS to integrate information from multiple multi-dimensional time-series data even when no single experiment provides full excitation. In the Markovian setting, physical constraints can be incorporated to enforce conservation laws. Numerical experiments on synthetic data demonstrate that NOMADS achieves substantially improved generalization accuracy compared to existing DMD-based methods even for noisy train data, and reproduces energy conservation in the Markovian case.

Haruka Ezoe, Kazuhiro Sato

To implement deep learning models on edge devices, model compression methods have been widely recognized as useful. However, it remains unclear which model compression methods are effective for Structured State Space Sequence (S4) models incorporating Diagonal State Space (DSS) layers, tailored for processing long-sequence data. In this paper, we propose to use the balanced truncation, a prevalent model reduction technique in control theory, applied specifically to DSS layers in pre-trained S4 model as a novel model compression method. Moreover, we propose using the reduced model parameters obtained by the balanced truncation as initial parameters of S4 models with DSS layers during the main training process. Numerical experiments demonstrate that our trained models combined with the balanced truncation surpass conventionally trained models with Skew-HiPPO initialization in accuracy, even with fewer parameters. Furthermore, our observations reveal a positive correlation: higher accuracy in the original model consistently leads to increased accuracy in models trained using our model compression method, suggesting that our approach effectively leverages the strengths of the original model.

Hiroki Sakamoto, Kazuhiro Sato

Deep learning models incorporating linear SSMs have gained attention for capturing long-range dependencies in sequential data. However, their large parameter sizes pose challenges for deployment on resource-constrained devices. In this study, we propose an efficient parameter reduction method for these models by applying $H^{2}$ model order reduction techniques from control theory to their linear SSM components. In experiments, the LRA benchmark results show that the model compression based on our proposed method outperforms an existing method using the Balanced Truncation, while successfully reducing the number of parameters in the SSMs to $1/32$ without sacrificing the performance of the original models.

Yosuke Kajiura, Kazuhiro Sato

We develop a generalized resistance geometry based on Kron reduction and effective resistance for directed graphs, paralleling classical undirected graph theory. For strongly connected directed graphs, we prove a Fiedler--Bapat identity that links the resistance matrix and the Laplacian through the symmetrized pseudoinverse. This identity provides a canonical definition of the resistance curvature and resistance radius in the strongly connected directed setting. In the strongly connected weight-balanced case, it also implies that the operation of associating an undirected Laplacian with a directed Laplacian via the pseudoinverse of the symmetrized pseudoinverse commutes with Kron reduction. We further introduce a class of signed undirected Laplacians for which effective resistance defines a distance between nodes. We call this distance the generalized resistance metric and prove that it coincides with the class of strict negative type metrics. Within this framework, we investigate analytical and geometric properties of resistance curvature and resistance radius, characterize the maximum graph-variance problem, and generalize resistive embeddings. These results place signed undirected resistance geometry on a footing parallel to the classical unsigned undirected theory and provide a unified perspective on model reduction, graph variance, and resistance-based embedding.

Hiroki Sakamoto, Kazuhiro Sato

We study deep state-space models (Deep SSMs) that contain linear-quadratic-output (LQO) systems as internal blocks and present a compression method with a provable output error guarantee. We first derive an upper bound on the output error between two Deep SSMs and show that the bound can be expressed via the $h^2$-error norms between the layerwise LQO systems, thereby providing a theoretical justification for existing model order reduction (MOR)-based compression. Building on this bound, we formulate an optimization problem in terms of the $h^2$-error norm and develop a gradient-based MOR method. On the IMDb task from the Long Range Arena benchmark, we demonstrate that our compression method achieves strong performance. Moreover, unlike prior approaches, we reduce roughly 80% of trainable parameters without retraining, with only a 4-5% performance drop.