[Paper Review] Unifying Theorems for Subspace Identification and Dynamic Mode Decomposition

Sungho Shin, Qiugang Lu+1 more|arXiv (Cornell University)|Mar 16, 2020

Model Reduction and Neural Networks20 references4 citations

TL;DR

This paper establishes a theoretical equivalence between subspace identification (SID) and dynamic mode decomposition (DMD) for autonomous dynamical systems, proving that their underlying optimization problems are identical under perfect data conditions. It proposes the SID-DMD algorithm, which leverages this equivalence to deliver a provably optimal, SVD-based state-space model directly from output data, enabling accurate and efficient system identification from high-dimensional data such as video sequences.

ABSTRACT

This paper presents unifying results for subspace identification (SID) and dynamic mode decomposition (DMD) for autonomous dynamical systems. We observe that SID seeks to solve an optimization problem to estimate an extended observability matrix and a state sequence that minimizes the prediction error for the state-space model. Moreover, we observe that DMD seeks to solve a rank-constrained matrix regression problem that minimizes the prediction error of an extended autoregressive model. We prove that existence conditions for perfect (error-free) state-space and low-rank extended autoregressive models are equivalent and that the SID and DMD optimization problems are equivalent. We exploit these results to propose a SID-DMD algorithm that delivers a provably optimal model and that is easy to implement. We demonstrate our developments using a case study that aims to build dynamical models directly from video data.

Motivation & Objective

• To establish a theoretical equivalence between subspace identification (SID) and dynamic mode decomposition (DMD) for autonomous systems.
• To resolve the lack of optimality guarantees in traditional SID methods by framing them as optimization problems.
• To develop a unified, provably optimal algorithm (SID-DMD) that leverages the equivalence for direct state-space model identification from output data.
• To demonstrate the method’s effectiveness on spatiotemporal data from liquid crystal sensing experiments.

Proposed method

• Formulates SID as a low-rank matrix regression problem minimizing prediction error in the state-space model via extended observability matrix estimation.
• Reformulates DMD as a rank-constrained matrix regression minimizing prediction error in an extended autoregressive model.
• Proves that the existence conditions for error-free SID and DMD models are equivalent, and that the two optimization problems are mathematically identical.
• Develops the SID-DMD algorithm: first solving a rank-constrained SVD-based regression, then estimating system matrices A and C from the solution, and finally extracting dynamic modes via eigendecomposition of A.
• Uses economic SVD and truncated SVD to efficiently compute the state sequence and extended observability matrix, ensuring numerical stability and low computational cost.
• Applies the method to video data by constructing delay-embedded Hankel matrices from time-series image frames, enabling identification of spatiotemporal dynamics.

Experimental results

Research questions

• RQ1 Are the optimization problems underlying SID and DMD mathematically equivalent under perfect data conditions?
• RQ2 Can a unified algorithm be derived that inherits the optimality of SID and the scalability of DMD?
• RQ3 Does the proposed SID-DMD method yield a provably optimal state-space model when the true system is low-rank and autonomous?
• RQ4 Can the unified framework accurately extract dynamic modes and associated timescales from high-dimensional video data?

Key findings

• The existence conditions for perfect SID and DMD models are equivalent, meaning that if a low-rank state-space model fits the data exactly, so does the corresponding DMD model.
• The SID and DMD optimization problems are mathematically equivalent, proving that both methods minimize the same prediction error under the same constraints.
• The proposed SID-DMD algorithm produces a provably optimal state-space model by solving a single rank-constrained regression problem via SVD, avoiding suboptimal truncation heuristics used in standard DMD.
• In the case study, the algorithm successfully identified three modes (one real, one conjugate pair) for both DMMP and water exposure in liquid crystals, revealing distinct dynamics: DMMP induced faster, more uniform transitions with shorter oscillation periods.
• The spatial modes for DMMP showed a boundary-to-center propagation pattern, indicating a more homogeneous diffusion process, while water exhibited less uniform spatial structures.
• The temporal trends revealed that DMMP’s real mode decayed more rapidly and its complex modes oscillated faster, confirming faster response dynamics observed visually in the video data.